b.ed science education - allama iqbal open university
TRANSCRIPT
i
B.Ed Science Education
PHYSICS-II FOR B.ED
Course Code: 6442 Units: 1–9
SCIENCE EDUCATION DEPARTMENT
FACULTY OF EDUCATION
ALLAMA IQBAL OPEN UNIVERSITY, ISLAMABAD
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(All Rights reserved with the publisher)
Edition .............................................. 1st
Year of Printing ............................... 2019
Quantity ........................................... 1000
Publisher .......................................... Allama Iqbal Open University, Islamabad
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COURSE TEAM
Chairman Course Team Prof. Dr. Nasir Mahmood
Course Development Coordinator Arshad Mehmood Qamar
Writers: 1. Hanifa Obaid
2. Dr. Amjad Imran
3. Imran Amin Asim
4. Arshad Mehmood Qamar
5. Mukhtar Adarsh
Reviewers: 1. Hanifa Obaid
2. Dr. Amjad Imran
3. Imran Amin Asim
4. Arshad Mehmood Qamar
5. Mukhtar Adarsh
Composer: Rehan Yaqoob
Editor: Fazal Karim
Layout: Asrar ul Haque Malik
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FORWARD
This course has been designed for fulfilling the content expertise of prospectus
teachers who will be enrolled in B.Ed 4years or B.Ed 2.5 Years in Allama Iqbal
Open University. This book is very useful for making up the need of the advance
content for students and teachers. This book will be helpful to reduce the
controversy that what type of knowledge, skills and values Science teachers need
in the subject of Physics. Some teachers need more content knowledge, whereas
some science teachers want to enhance the knowledge. Further Scientific
knowledge is expanding at very high speed. This is the era of scientific innovation
and creations. Innovations and creations need skillful technologies. Allama Iqbal
Open University and Science Education Department has promised to maintain the
quality and acceptability. This book is one of those series of books which will
enable the teachers to cope with changing needs of the society and students.
This book is not written by a single author but a group of authors having vast
experience in the field of Physics. Dr. Farkhunda Rashid Ch., Ms. Saima Kashif,
Arshad Mehmood Qamar, Dr. Muhammad Tanveer Afzal from Science Education
Department and writers Like Dr. Amjad , Ms. Hanifa Ubaid and other team
worked very hard to make this book beneficial for students. Other students who
are not students of this University can also get benefit from this text.
The focus of this book is to provide the students with best knowledge, skills and
content in the subject of biological sciences. With the help of this book science
students can explore the Phenomenons related to physical world. Keeping in
view the qualitative aspect of education and an increasing demand of science
teachers, stress is laid upon Physics content as well as strengthening their
professional skills and knowledge. The elements of motivation and love are also
considered. We welcome suggestions and comments for improvements from the
readers, teachers and public at large for the improvement of this course.
Prof. Dr. Nasir Mahmood
Chairman
Science Education Department
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PREFACE
Though there is lot of books available in market, but there is no book which
fulfills the requirements of University approved outlines. Some cover one area of
content while other covers another area. In this way there would be a lot of
financial burden and dispersed focus.
Further AIOU has its own requirement either to provide compiled material or
textbook. This book is one of those series for coverage of content area
requirement for B.Ed 4 Year and B.Ed 2.5 Year in the field of Physics.
This book is written as per prescribed procedure of book development. After
approval of content from all statuary bodies, approval for starting development of
this book was sought. Then selection of writers and reviewers was completed.
Time and again reminders to Unit writers and telephonic conversation were done
with the writers to expedite the process of writing and review. In spite of very
tedious work of writers and reviewers, coordinators had to look into everything
i.e. Format, self assessment exercises, alignment of the content and addition of
some essential things and removal of irreverent things.
Great stress has been laid in making the course to facilitate prospectus, in service
and pre-service teachers for content knowledge regarding Physics. The course is
equipped with illustrations for better understanding of the reader. Each unit is
equipped with necessary illustrations, activities and self assessment exercises.
AIOU hope that this book will prove best for the content knowledge regarding
Physics.
Arshad Mehmood Qamar
Course Development Coordinators
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ACKNOWLEDGEMENTS
Allama Iqbal Open University and the Course Development Coordinator along
with course development team are grateful to the writers and publishers of
Physics books for adopting their books and materials, internet for providing useful
information regarding Physics, and reference materials for the development of the
course of Physics-II for B.Ed Science Education.
All are specially acknowledged whose information and material have been quoted
in the course that Allama Iqbal Open University is a non- commercial educational
University in Pakistan which is providing educational facilities to under-
privileged remote rural areas through distance and non-formal mode.
It is a matter of pleasure for department of science education and AIOU to
acknowledge all those whose efforts and hard work make it possible to frame
Contents of this book. Committee of courses tried her best to make necessary
changes and then approved the contents of this course.
Highly acknowledged members of CoC Prof. Dr. Rizwan Akram Rana, Dr.
Muhammad Idrees, Dr. Hafiz Athar Khan, Dr. Fazal Ur Rahman, Dr. Muhammad
Samiullah, Dr. Farkhunda Rashid Ch. and Arshad Mehmood Qamar.
My special thanks to Mr. Arshad Mehmood Qamar Course Development
Coordinator who worked with the same team whose approval was got by Dr.
Farkhunda Rashid Ch. Course team include; Amjad Imran, Mr. Imran Amin
Asim, Hanifa Ubaid, Arshad Mehmood and other members of the team
acknowledge the writers for writing the units efficiently.
I also acknowledge the team of CP, editor, lay out designer, composer and
producer for giving their input to make this book more beautiful. Our PPU team is
very cooperative and helpful for publishing the book. Finally I acknowledged all
those who in one way or the other put their efforts for completion of this task.
Prof. Dr. Nasir Mehmood
Chairman/ Dean
Faculty of Education
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INTRODUCTION The study of waves and oscillations provides basis for quantum mechanics. Much
of the quantum deals with the study of the “wave function” which describes the
wave nature of objects. However the connection of the field of waves and
oscillations to that of quantum mechanics is much deeper. Therefore this course
consists of three parts. In first part wave and oscillations have been discussed. The
second part contains the concepts of special theory of Relativity, light,
interference, diffraction and polarization. In third part the content of quantum
physics is described according to the AIOU format.
OBJECTIVES After completing this course, you will be able to:
understand the mathematics of waves and oscillations and how that applies
to position, velocity and other quantities.
discuss the relationship between SHM and wave motion, and make
calculations of motion of systems of masses connected by springs.
make calculations on simple properties of wave motion , including wave
packets, phase velocity, group velocity and the propagation of waves in one
dimention, dispersion of waves.
sketch and describe standing waves, especially on strings and in pipes with
various boundary conditions.
apply the principle of interference with diffraction to waves that pass
through a single or double slit to through multiple slits through a diffraction
grading.
describe different polarization states.
derive Lorenentz transformation and explain its consequences.
describe the consequences of theory of relativity.
examine properties of waves through experiments.
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CONTENTS
Unit 1: Harmonic Oscillations ............................................................1
Unit 2: Waves and Physical Media ...................................................25
Unit 3: Special Theory of Relativity .................................................39
Unit 4: Light ......................................................................................57
Unit 5: Interference ...........................................................................67
Unit 6: Diffraction .............................................................................77
Unit 7: Polarization ...........................................................................89
Unit 8: Origin of Quantum Theory .................................................105
Unit 9: Wave nature of Matter ........................................................119
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UNIT-1
HARMONIC OSCILLATIONS
Written by: Imran Amin
Reviewed by: Arshad Mehmood Qamar
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CONTENTS
Introduction ....................................................................................................... 3
Objectives ......................................................................................................... 3
1.1 Oscillations ........................................................................................... 4
1.2 Simple Harmonic Oscillations .............................................................. 4
1.3 Obtaining and solving the Basic Equations of Motion ......................... 5
1.4 Longitudinal and Transverse Oscillations ............................................ 7
1.5 Energy Consideration in SHM .............................................................. 10
1.6 Applications of SHM ............................................................................ 12
1.7 Simple Harmonic Motion and Uniform Circular Motion ..................... 17
1.8 Combination of Harmonic Motion........................................................ 18
1.9 Lissajous Patterns.................................................................................. 20
1.10 Damped Oscillation .............................................................................. 21
1.11 Forced Oscillation and Resonance ........................................................ 22
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Introduction
You see most examples of repeated motion in your daily life. If a body repeatedly returns
to its original position, we call its motion repetitive. Main examples of repetitive motion
of the human body are heart beat and breathing. There are so many bodies which move in
a repetitive way e.g a rocking chair, a swing and a simple clock pendulum. Ancient
Scientists had observed the motion of sun and the phases of moon which was firstly
observed as repetitive motion. In musical instruments, repetitive motion is the physical
basis of all the stringed. Can you tell me the common properties of these given examples
of repetitive motion? In this chapter we will discuss the physical characteristics of
oscillation/repetitive motion, and we will develop techniques that can be used to analyze
this motion quantitatively.
Objectives
After studying this Unit, the students will be able to;
i. understand Simple Harmonic Oscillation (SHO).
ii. describe different types of SHO.
iii. explain Longitudinal and Transverse Oscillation.
iv. identify oscillatory motion by using different techniques.
v. elaborate the SHM and Uniform Circular Motion
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1.1 Oscillations
A body is said to be vibrating if it moves back and forth or to and fro about a fix point.
Vibration is also known as oscillation. A special kind of vibratory or oscillatory motion is
called the simple harmonic motion (SHM). In other words, oscillation is the repetitive
variation, typically in time, of some measure about a central value or between two or
more different states. It is usually a back and forth motion over a central neutral point,
created by changes in energy. In a pendulum-driven clock, for example, the oscillation is
the back and forth movement of the pendulum.
1.2 Simple Harmonic Oscillations In the following sections we will discuss
about the Simple Harmonic Oscillation.
According to the Hook’s Law, the motion of
a particle moving along a straight line with
an acceleration whose direction is always
towards a fixed point on the line and whose
magnitude is proportional to the distance
from the fixed point is called simple
harmonic oscillation or simple harmonic
motion (SHM).
Consider a body of mass is attached to a spring of spring constant is executing the
oscillatory motion. The restoring force acting on the object can be find out by using
expression:
(1.1)
The acceleration produced in the body of
mass due to restoring force can be calculated
using second law of motion.
(1.2)
Comparing equation (1.1) and (1.2), we get:
where k and m are constant so we get
(1.3)
This expression shows that the acceleration of
the oscillating object is directly proportional
to displacement and is directed towards the
mean position. Thus the object attached to
horizontal mass spring system executes simple harmonic motion.
Figure 1.1 Mass attach to spring
Fig: 1.2 K.E & P.E
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The negative sign in Eq. 1.1 means that the force exerted by the spring is always directed
opposite to the displacement of the mass. Because the spring force always acts towards
the mean position, it is sometimes called a restoring force.
A restoring force always pushes or pulls the object performing oscillatory motion
towards the mean position.
Here we will discuss some important features of SHM, such as:
i. A body executing SHM always vibrates about a fixed position.
ii. Its acceleration is always directed towards the mean position.
iii. The magnitude of acceleration is always directly proportional to its displacement
from the mean position i.e., acceleration will be zero at the mean position while it
will be maximum at the extreme positions.
iv. Its velocity is maximum at the mean position and zero at the extreme positions.
Now we discuss different terms which characterize simple harmonic motion.
Vibration: One complete round trip of a vibrating body about its mean
position is called one vibration.
Time Period (T ): The time taken by a vibrating body to complete one vibration is
called time period.
Frequency ( f ): The number of vibrations or cycles of a vibrating body in one
second is called its frequency. It is reciprocal of time period i.e.,
f = 1/T
Amplitude (A): The maximum displacement of a vibrating body on either side
from its mean position is called its amplitude.
1.3 Obtaining and Solving the Basic Equations of Motion
The general equation for simple harmonic motion along the x-axis results from a straight
forward application of Newton's second law to a particle of mass m acted on by a force:
F = -kx
where x is the displacement from
equilibrium and k is called the
spring constant.
Since the acceleration:
a = dv/dt = d2x/dt
2 (1.4)
Newton's second law becomes:
-kx = m d2x/dt
2 (1.5)
which is called a second-order
differential equation because it
contains a second derivative.
We can combine the constants k and m by making the substitution:
k/m = 2
(1.6)
and rewrite this equation as:
Figure 1.3
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d2x/dt
2 = -
2x (1.7)
Above equation (1.7) is not like an algebraic equation for which certain constant values
of x satisfy the equality.
The solution of Eq. (1.7) is a function of time. Although the function
x(t) = A cos( t + ) = A cos( ) cos( t) - A sin( ) sin( t) (1.8)
can be thought of simply as being arrived at by a very clever guess, it can be shown (by
advanced mathematical techniques) to be the most general possible solution of Eq. (1.7).
Equation (1.8) can also be written in terms of two new constants B and C as
x(t) = B cos( t) + C sin( t) (1.9)
The velocity is then
v(t) = dx/dt = - B sin( t) + C cos( t) (1.10)
These last two equations are especially helpful.
For instance, if you are told that a particle begins its simple harmonic motion from rest at
the point x0, you know that x(0) = x0 and v(0) = 0; hence, since cos(0) = 1 and sin(0) = 0
you immediately have:
B = x0 and C = 0.
If the particle starts at the origin (x=0) with velocity v0, then you can conclude that:
B = 0 and C = v0.
Look at the equations and check these results for yourself!
If you have a more complicated case in which the particle starts at x0 with velocity v0,
then you can find B and C yourself, using the same method. Try it!
Once you have found B and C, you can then find A and .
Familiarize yourself with the symbols used to identify the parameters of harmonic
motion, and recognize that different texts may use different symbols.
For example, you may see or (instead of ) used to represent the phase constant.
The phase constant determines the initial conditions (displacement and velocity) of the
motion. The significance of the angular nature of the phase constant should become
somewhat clearer in the next section when we examine the analogy between harmonic
motion and uniform circular motion. For now, notice that since is a constant, its value is
arbitrary in the general solution of Eq. (1.7). Even when the initial conditions of the
motion (at t = 0) are specified, is only determined to within an integral multiple of 2. That
is, if
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x = A cos( t + ) (1.11)
describes the motion, then so does
x = A cos( t + + 2n ) (1.12)
for any integer n= 1, 2,3, ....
The equation:
x = A sin( t + + /2) (1.13)
is an equally valid (in fact identical) solution as you can verify for yourself.
Be alert to the difference between frequency, f or v and angular frequency .
Both can have dimensions of s-1
, but the units of frequency are oscillations/second while
those of angular frequency are radians/second: they are related by:
= 2 f (1.14)
Self Assessment Questions
What is the relation between frequency and angular frequency?
When does a body execute SHM?
1.4 Longitudinal and Transverse Oscillations
When we think of the word "wave" we usually picture someone moving their hand back
and forth to say hello or maybe we think of a curling wall of water moving in from the
ocean to crash on the beach.
In physics, a wave is a disturbance that travels through space and matter
transferring energy from one place to another. When studying waves it's important to
remember that they transfer energy, not matter.
Waves can be divided into various categories depending on their characteristics. Below
we describe some of the different terms that scientists use to describe waves.
1.4.1 Mechanical Waves and Electromagnetic Waves All waves can be categorized as either mechanical or electromagnetic.
Mechanical waves are waves that require a medium. This means that they have to have
some sort of matter to travel through. These waves travel when molecules in the medium
collide with each other passing on energy. One example of a mechanical wave is sound.
Sound can travel through air, water, or solids, but it can't travel through a vacuum. It
needs the medium to help it travel. Other examples include water waves, seismic waves,
and waves traveling through a spring.
8
Electromagnetic waves are waves that can travel through a vacuum (empty space). They
don't need a medium or matter. They travel through electrical and magnetic fields that are
generated by charged particles. Examples of electromagnetic waves include light,
microwaves, radio waves, and X-rays.
Another way to describe a wave is by the direction that its disturbance is traveling.
Depending upon the direction of displacement of medium with respect to the direction of
the propagation of wave itself, mechanical waves may be classified as longitudinal and
transverse.
1.4.2 Longitudinal Oscillation Longitudinal waves can be produced on a spring (slinky) placed on a smooth floor or a
long bench. Fix one end of the slinky with a rigid support and hold the other end into
your hand. Now give it a regular push and pull quickly in the direction of its length A
series of disturbances in the form of waves will start moving along the length of the
slinky. Such a wave consists of regions called compressions, where the loops of the
spring are close together, alternating with regions called rarefactions (expansions),
where the loops are spaced apart. In the regions of compression, particles of the medium
are closer together while in the regions of rarefaction; particles of the medium are spaced
apart. The distance between two consecutive compressions is called wavelength. The
compressions and rarefactions move back and forth along the direction of motion of the
wave. Such a wave is called longitudinal wave and is defined as:
Consider a coil of spring capable of vibrating horizontally. Suppose an oscillating force F
is applied to its ends horizontally. The force will alternately stretch and compress the
spring, thereby sending a series of stretched regions (rarefactions) and compressions
down the spring. Such types of waves are also called the compression waves. These
Fig: 1.4 Longitudinal Waves
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compressions and rarefactions are represented by corresponding crests and troughs as
show in figure below:
Fig: 1.5 Longitudinal wave on a slinky
In longitudinal waves the particles of the medium move back and forth along the
direction of propagation of wave.
1.4.3 Transverse Oscillations We can produce transverse waves with the help of a slinky. Stretch out a slinky along a
smooth floor with one end fixed. Grasp the other end of the slinky and move it up and
down quickly. A wave in the form of alternate crests and troughs will start travelling
towards the fixed end. The crests are the highest points while the troughs are the lowest
points of the particles of the medium from the mean position. The distance between two
consecutive crests or troughs is called wavelength. The crests and troughs move
perpendicular to the direction of the wave. Therefore, transverse waves can be defined as:
Consider a string whose left end is connected to a blade that is set vibrating. The wave
travels towards right as the crest and trough in turn, replace each other. Every part of the
string, such as point P, oscillates vertically with simple harmonic motion.
Figure 1.6 Transverse wave on a slinky
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1.5 Energy Consideration in SHM
To produce a deformation in an object, we must do work. That is, whether you pluck a
guitar string or compress a car’s shock absorber, a force must be exerted through a
distance. If the only result is deformation, and no work goes into thermal, sound, or
kinetic energy, then all the work is initially stored in the deformed object as some form of
potential energy.
Consider the example of a block attached to a spring on a frictionless table, oscillating in
SHM. The force of the spring is a conservative force (which you studied in the chapter on
potential energy and conservation of energy), and we can define a potential energy for it.
This potential energy is the energy stored in the spring when the spring is extended or
compressed. In this case, the block oscillates in one dimension with the force of the
spring acting parallel to the motion:
When considering the energy stored in a spring, the equilibrium position, marked
as xi=0.00 m, is the position at which the energy stored in the spring is equal to zero.
When the spring is stretched or compressed a distance x, the potential energy stored in the
spring is
U =
k x
2
In a simple harmonic oscillator, the energy oscillates between kinetic energy of the
mass K.E =
mv
2 and potential energy U =
kx
2 stored in the spring. In the SHM of the
mass and spring system, there are no dissipative forces, so the total energy is the sum of
the potential energy and kinetic energy. In this section, we consider the conservation of
energy of the system. The concepts examined are valid for all simple harmonic
oscillators, including those where the gravitational force plays a role.
In the case of undamped SHM, the energy oscillates back and forth between kinetic and
potential, going completely from one form of energy to the other as the system oscillates.
So for the simple example of an object on a frictionless surface attached to a spring, the
motion starts with all of the energy stored in the spring as elastic potential energy. As
the object starts to move, the elastic potential energy is converted into kinetic energy,
becoming entirely kinetic energy at the equilibrium position. The energy is then
converted back into elastic potential energy by the spring as it is stretched or compressed.
The velocity becomes zero when the kinetic energy is completely converted, and this
cycle then repeats. Understanding the conservation of energy in these cycles will provide
extra insight here and in later applications of SHM, such as alternating circuits.
The transformation of energy in SHM for an object attached to a spring on a frictionless
surface. (a) When the mass is at the position x=+A , all the energy is stored as potential
energy in the spring U=
kA
2. The kinetic energy is equal to zero because the velocity of
the mass is zero. (b) As the mass moves toward x=−A, the mass crosses the position x=0.
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At this point, the spring is neither extended nor compressed, so the potential energy
stored in the spring is zero. At x=0, the total energy is all kinetic energy where K=
m(−vmax)2. (c) The mass continues to move until it reaches x=−A, where the mass stops
and starts moving toward x=+A. At the position x=−A, the total energy is stored as
potential energy in the compressed U=
k(−A)
2 and the kinetic energy is zero. (d) As the
mass passes through the position x=0, the kinetic energy is K=
mvmax
2 and the potential
energy stored in the spring is zero. (e) The mass returns to the position x = +A , where K
= 0 and U=
kA
2.
Self Assessment Question Tell whether or not these motions are examples of simple harmonic motion:
a) Ups and down motion of a leaf in water pond
b) Motion of a ceiling fan
c) Motion of hands of clock
d) Movement of honey bee.
e) Motion of a plucked string fixed at both its ends
Fig: 1.7 Simple Harmonic Motion
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1.6 Applications of SHM There are many applications of SHM in our daily life. Here we will discuss some of
them.
Clock Depending on the type of clock and the era that it was
made, either a large pendulum or a vibrating quartz
crystal will keep time. This motion of the pendulum or
crystal must be periodic, in order to ensure that the time
shown will be accurate.
The oscillators have a constant period because they are in
Simple Harmonic Motion. This allows them to keep time
accurately. This accurate time is more than just for
convenience. It is a necessity when navigating using
celestial marks like the Sun or stars because the position of
the Sun and stars in the sky changes with the time of day.
It is also important that the period is independent of the
amplitude of oscillation. This is because the conditions
in a grandfather clock can vary, and affect the
amplitude of oscillation. No matter how the amplitude varies, the time shown is always
constant.
Car Shock Absorbers Springs attached to the wheels of a car help ensure a
smoother ride for the passengers. When the car hits a
bump in the road, the wheels of the car rise up. If the car
did not have shock absorbers, the whole car will be
thrown up and down, causing an extremely
uncomfortable and possibly dangerous ride.
When there are springs in the car, the wheel will rise,
compressing the spring while the car body remains
relatively stationary. The restoring force on the
compressed spring will push the car wheel back into
place.
However, when the spring is put into SHM by the bump
in the road, it continues to bounce up and down even
though the unevenness that triggered the motion is gone.
This is why cars contain shock absorbers, which are usually pistons filled with a viscous
liquid, to damp the oscillations quickly. This is an example of over damping.
Figure 1.8 Pendulum Clock
Figure 1.9 Car Shock Absorbers
13
Musical Instruments Sound itself is produced from oscillations of the air. In a string instrument such as a
violin or a guitar, bowing or plucking the string provides the force needed to make the
string oscillate and produce sound. In a wind instrument like a trumpet, the vibrations are
Fig: 1.10 Musical Instrument
caused by the player's lips while the sound is caused by exciting the air molecules by
blowing across the opening in a flute.
In a percussion instrument like the triangle, the vibrations occur when the instrument is
struck. The vibration produced in the string, column of air or body of the instrument
causes standing waves to be formed, which produces sound.
Self Assessment Questions 1. Enlist some other applications of SHM.
2. Write names of some musical instruments.
1.6.1 Physical Pendulum Any object can oscillate like a pendulum. Consider
a coffee mug hanging on a hook in the pantry. If
the mug gets knocked, it oscillates back and forth
like a pendulum until the oscillations die out. We
have described a simple pendulum as a point mass
and a string. A physical pendulum is any object
whose oscillations is similar to those of the simple
pendulum, but cannot be modeled as a
point mass on a string, and the mass
distribution must be included into the
equation of motion.
As for the simple pendulum, the restoring force of
the physical pendulum is the force of gravity. With the simple pendulum, the force of
Fig: 1.11 Physical Pendulum
14
gravity acts on the center of the pendulum bob. In the case of the physical pendulum, the
force of gravity acts on the center of mass (CM) of an object. The object oscillates about
a point O. Consider an object of a generic shape as shown in Figure
When a physical pendulum is hanging from a point but is free to rotate, it rotates because
of the torque applied at the CM, produced by the component of the object’s weight that
acts tangent to the motion of the CM. Taking the counterclockwise direction to be
positive, the component of the gravitational force that acts tangent to the motion is −mg
sin θ . The minus sign is the result of the restoring force acting in the opposite direction
of the increasing angle. Recall that the torque is equal to τ =r ×F . The magnitude of
the torque is equal to the length of the radius arm times the tangential component of the
force applied, | τ | = rFsin θ . Here, the length L of the radius arm is the distance between
the point of rotation and the CM. To analyze the motion, start with the net torque. Like
the simple pendulum, consider only small angles so that sin θ ≈ θ . Recall from Fixed-
Axis Rotation on rotation that the net torque is equal to the moment of inertia I = ∫ r2 dm
times the angular acceleration α , where
I α = τnet = L (−mg) sinθ (1.15)
Using the small angle approximation and rearranging:
I α = −L(mg)θ
I θ
= −L(mg)θ
θ
= −
θ
Once again, the equation says that the second time derivative of the position (in this case,
the angle) equals minus a constant (−mgL/I) times the position. The solution is
where Θ is the maximum angular displacement. The angular frequency is
The period is therefore
Note that for a simple pendulum, the moment of inertia is I = ∫ r2dm = mL2 and the
period reduces to
15
1.6.2 Torsional Pendulum A torsional pendulum consists of a rigid body
suspended by a light wire or spring (Figure 8).
When the body is twisted some small maximum
angle (Θ) and released from rest, the body
oscillates between (θ = + Θ) and (θ = − Θ). The
restoring torque is supplied by the shearing of the
string or wire.
The restoring torque can be modeled as being
proportional to the angle:
τ = − κ θ
The variable kappa (κ) is known as the torsion constant of the wire or string. The minus
sign shows that the restoring torque acts in the opposite direction to increasing angular
displacement. The net torque is equal to the moment of inertia times the angular
acceleration.
This equation says that the second time
derivative of the position (in this case, the angle)
equals a negative constant times the position.
This looks very similar to the equation of
motion for the SHM d2x/dt
2 = − kx/m , where
the period was found to be T = 2π√m/k.
Therefore, the period of the torsional pendulum
can be found using
T=2π√I/κ.
The units for the torsion constant are [κ] = N
•.m = (kg • m/s2)m = kg • m
2/s
2 and the units for
the moment of inertial are [I] = kg • m2, which
show that the unit for the period is the second.
1.6.3 Simple Pendulum A simple pendulum is defined to have a point mass, also known as the pendulum bob,
which is suspended from a string of length L with negligible mass (Figure 9). Here, the
only forces acting on the bob are the force of gravity (i.e., the weight of the bob) and
tension from the string. The mass of the string is assumed to be negligible as compared to
the mass of the bob.
Fig: 1.12 Torsional Pendulum
Fig: 1.13 Simple Pendulum
16
Consider the torque on the pendulum. The force providing the restoring torque is the
component of the weight of the pendulum bob that acts along the arc length. The torque
is the length of the string L times the component of the net forces that is perpendicular to
the radius of the arc. The minus sign indicates the torque acts in the opposite direction of
the angular displacement:
The solution to this differential equation involves advanced calculus, and is beyond the
scope of this text. But note that for small angles (less than 15°), sin θ and θ differ by less
than 1%, so we can use the small angle approximation sin θ ≈ θ. The angle θθ describes
the position of the pendulum. Using the small angle approximation gives an approximate
solution for small angles,
Because this equation has the same form as the equation for SHM, the solution is easy to
find. The angular frequency is
and the period is
The period of a simple pendulum depends on its length and the acceleration due to
gravity. The period is completely independent of other factors, such as mass and the
maximum displacement. As with simple harmonic oscillators, the period T for a
pendulum is nearly independent of amplitude, especially if θθ is less than about 15°.
Even simple pendulum clocks can be finely adjusted and remain accurate.
17
1.7 Simple Harmonic Motion and Uniform Circular Motion
If we tie a stone to the end of a string and move it with a constant angular speed in a
horizontal plane about fixed point, the stone would perform a uniform circular motion in
the plane. If we observe the stone sideways, the stone will appear to perform a to and fro
motion along the horizontal line with the other end of the string as the midpoint.
Similarly, the projection of the motion or the shadow of the stone would appear to
perform a to and fro motion perpendicular to the plane of the circle. Similar case was
observed by Galileo, who discovered the four principle moons of the planet Jupiter
moved back and forth relative to the planet, executing a simple harmonic motion.
At the right is an object going in a circle on a rotating
turntable. Just behind the object is a screen where the
shadow of the object can be seen. The shadow moves
back and forth as the object goes in a circle. The
shadow appears to be in SHM. At the right is a sketch
of the object in uniform circular motion. It has a
centripetal acceleration, a tangential velocity, and a
position vector all shown. If we redraw the three
vectors with their tails at the origin we can imagine all
three spinning as the object rotates. Finding the x-
components of each – the position, velocity, and
acceleration of the shadow
x = r cosθ , vx = −vsinθ and ax = −acosθ .
The tangential velocity is related to this angular
velocity,
v = ωr.
Also, the centripetal acceleration is related to the
angular velocity,
a = v2 r = (ωr)
2 r ⇒ a = ω
2 r.
Substituting, we get,
x = r cosθ , vx = −ωrsinθ , and ax = −ω2 r cosθ .
Notice, just like SHM we have ax = −ω2 x .
The angle θ changes with time. We can write this using the definition of angular
frequency,
ω ≡ dθ dt ⇒ dθ ∫ = ωdt ∫ ⇒ θ = ωt + δ.
Fig: 1.14 Circular Motion
18
Now we see another way of looking at the phase angle, δ, as just an integration constant.
Finally, we can write the x-components of the, position, velocity and acceleration for the
oscillating shadow as a function of time.
These are the same as the SHM equations of motion with A instead of r. The x
component of the motion of an object in uniform circular motion is SHM. That explains
why we keep talking about angular frequencies!
1.8 Combination of Harmonic Motion
Consider the superposition, or interference of two simple harmonic motions, that produce
a displacement of the particle along the same line. Suppose that both have the same
frequency. The displacement time functions are given by:
Fig: 1.15 Related
to Example
EXAMPLE: A 500g mass rests in equilibrium at the end of a
horizontal spring with spring constant 9.80N/m. The mass is
given a sharp kick resulting in an initial velocity of 0.443m/s to
the right. (a) Sketch the initial position, velocity, and
acceleration vectors as if the object were in circular motion.
Find (b) the location of the equivalent object in circular motion,
(c) the phase angle, and (d)the equation for v(t).
Given: k = 9.80N/m, m = 0.500kg, v(0) = 0.443m/s, and x(0) = 0.
Find: r = ? , v = ? , a = ? , d = ?, and x(t)=?
a) We are given that the velocity vector is to the right and the
initial position is zero. Therefore, the x-component of the
velocity must be at a maximum and point to the right. The
xcomponent of the position and acceleration vectors must be
zero. The acceleration must point toward the center of the
circle and the position must point outward. The answer then is
in the sketch at the upper right.
b) The equivalent object in circular motion with the vectors
pointing the right direction must be as shown at the right.
c) Looking at the circle, the phase angle must be 270˚ or 3π/2 .
d) Using the appropriate equation of motion for circular motion,
the equation for the position as a function of time is v(t) = −vo
sin(ωt + δ ). The angular frequency for a spring is,
ω = (k/ m) = (9.8/ 5 = 4.43 rad s . So, v(t) =
−0.443sin(4.43t + 3π/ 2 ).
Note this results in v(0) = +0.443m/s as required
19
The resulting displacement is given by:
So the result is a harmonic oscillation with the same frequency:
Where
solving this system for A and ϕ :
The amplitude of the resultant oscillation depends on the amplitudes of the two original
amplitudes and on the difference of the two original initial phases. Let us consider some
important special cases: If ϕ1 =ϕ2 ⇒ ϕ1- ϕ2 = 0 . The two motions are in phase. The
amplitude of the interference is:
Hence the two simple harmonic motions interfere by reinforcement because their
amplitude adds. If ϕ1- ϕ2 = π, then two oscillations are in opposite phase, and the
amplitude:
In this case we say, that the two simple harmonic motions are in opposition, and the two
simple harmonic motions interfere by attenuation, because their amplitudes subtract. If
the two amplitudes are equal A1 = A2 , they completely cancel each other.
20
1.9 Lissajous Patterns A Lissajous curve is also known as Lissajous figure
which is the graph of a system of parametric equations. Let
each vibration be a simple harmonic motion represented by
a sinusoidal wave; let a and b denote the amplitudes, ω1
and ω2 the angular frequencies (in radians per second), φ1
and φ2 the phases, and t the time. We then have
x = a sin (ω1t + φ1)‘ ; y = b sin (ω2t + φ2)
As time progresses, the point P whose coordinates are •x;
y‘ will trace a curve whose equation can be found by
eliminating t between equations (1). Since the two
equations contain six parameters,1 the curve is usually
quite complicated, except in some special cases. For
example, if ω1 = ω2 and φ1 = φ2, we have
x = a sin (ωt + φ); y = b sin (ωt + φ)‘
where we have dropped the subscripts under the
parameters. To eliminate t between these equations, we
note that x/a = y/b and thus y =xb/a, the equation of a
straight line. Similarly, for ω1 = ω2 and a phase difference
of π we get the line y= −xb/a. For ω1 = ω2 and a phase
difference of π/2, we have (taking φ1 to be zero)
x = a sin ωt; y = b sin(ωt + π/2) = b cos ωt
Dividing the first equation by a and the second by b,
squaring the results, and adding, we get
x2/a
2 + y
2/b
2 = 1;
which represents an ellipse with axes along the x- and y-
axis (if in addition a = b, the ellipse becomes a circle). For
an arbitrary phase difference the curve will be a tilted
ellipse, of which the previous examples are special cases
(the lines y = ±•xb/a being degenerate ellipses). If we let the phase difference vary
continuously, the ellipse will slowly change its orientation and shape, passing (in the case
a = b) from the circle x2+y
2=1 to the lines y = ±x . If the frequencies are unequal, the
curve is much more complex. For example, when ω2 = 2ω1 (musically speaking, when the
two vibrations are an octave apart), we have
x = a sin ωt; y = b sin (2ωt + φ)
Fig: 1.16 Lissajous Patterns
21
where we again dropped the subscripts and took φ1 = 0. The kind of curve we get now
depends on φ. For φ = π/2 we have
x = a sin ωt; y = b sin (2ωt + π/2) = b cos 2ωt
Using the identity cos2u = 1 − 2sin2 u and eliminating t between the two equations results
in y = b’ 1 − 2(x/a)2. This equation represents a parabola along which the point P moves
back and forth as time progresses. For other values of φ the curve may have a closed
shape . One other observation is worth mentioning. As long as the frequency ratio ω1/ω2
is a rational number, the curve—no matter how complex—will eventually repeat itself,
causing the motion to be periodic.2 But if ω1/ω2 is irrational, the point P will never
retrace its own path, resulting in a non-periodic motion. However, as time progresses the
curve will gradually fill the rectangle bounded by the lines
x = ±a; y = ±b
1.10 Damped Oscillation
The oscillatory motions we have considered so far have been for ideal systems — that is,
systems that oscillate indefinitely under the action of a linear restoring force. In many
real systems, dissipative forces, such as friction, retard the motion. Consequently, the
mechanical energy of the system diminishes in time, and the motion is said to be damped.
One common type of retarding force is the one discussed in Section 1.2, where the force
is proportional to the speed of the moving object and acts in the direction opposite the
motion. This retarding force is often observed when an object moves through air, for
instance. Because the retarding force can be expressed as R= -bv (where b is a constant
called the damping coefficient) and the restoring force of the system is kx, we can write
Newton’s second law as:
The solution of this equation requires mathematics that may not be familiar to you yet;
we simply state it here without proof. When the retarding force is small compared with
the maximum restoring force—that is, when b is small— hence the solution is:
where the angular frequency of oscillation is
We see that when the retarding force is much smaller than the restoring force, the
oscillatory character of the motion is preserved but the amplitude decreases in time, with
22
the result that the motion ultimately ceases. Any system that behaves in this way is
known as a damped oscillator.
Information: The oscillations of a system in the presence of some resistive force
are damped oscillations.
Shock absorbers in automobiles are one practical application of damped motion. A shock
absorber consists of a piston moving through a liquid such as oil. The upper part of the
shock absorber is firmly attached to the body of the car. When the car travels over a
bump on the road, the car may vibrate violently. The shock absorbers damp these
vibrations and convert their energy into heat energy of the oil.
1.11 Forced Oscillation and Resonance
It is possible to compensate for energy loss in a damped system by applying an external
force that does positive work on the system. At any instant, energy can be put into the
system by an applied force that acts in the direction of motion of the oscillator. For
example, a child on a swing can be kept in motion by appropriately timed pushes. The
amplitude of motion remains constant if the energy input per cycle exactly equals the
energy lost as a result of damping. Any motion of this type is called forced oscillation. A
common example of a forced oscillator is a damped oscillator driven by an external force
that varies periodically, such as where is the angular frequency of
the periodic force and Fext is a constant.
After a sufficiently long period of time, when the energy input per cycle equals the
energy lost per cycle, a steady-state condition is reached in which the oscillations proceed
with constant amplitude. At this time, when the system is in a steady state:
The external agent provides the necessary energy to overcome the losses due to the
retarding force. Note that the system oscillates at the angular frequency ω of the driving
force. For small damping, the amplitude becomes very large when the frequency of the
driving force is near the natural frequency of oscillation. The dramatic increase in
amplitude near the natural frequency ω0 is called resonance, and for this reason ω0 is
sometimes called the resonance frequency of the system.
23
When the applied force F is in phase with the velocity, the rate at which work is done on
the oscillator by F equals the dot product F and v. Remember that “rate at which work is
done” is the definition of power. Because the product F v is a maximum when F and v are
in phase, we conclude that at resonance the applied force is in phase with the velocity and
that the power transferred to the oscillator is a maximum.
Self Assessment Questions
Is a bouncing ball an example of simple harmonic motion?
Is the daily movement of a student from home to school and back simple harmonic
motion? Why or why not?
24
25
UNIT-2
WAVE AND PHYSICAL MEDIA
Written by: Hanifa Obaid
Reviewed by: Dr. Amjad Imran
26
CONTENTS
Introduction ....................................................................................................... 27
Objectives ......................................................................................................... 27
2.1 Mechanical Waves ................................................................................ 28
2.2 Types of Waves..................................................................................... 28
2.3 Travelling Waves .................................................................................. 29
2.4 Phase Velocity of Travelling Wave ...................................................... 30
2.5 Waves Sinusoidal Waves ...................................................................... 30
2.6 Group Speed and Dispersion ................................................................ 31
2.7 Wave Equation ...................................................................................... 33
2.8 Power and Intensity in Wave Motion ................................................... 34
2.9 The Principle of Superposition ............................................................. 35
2.10 Interfernace of Standing Waves ............................................................ 35
2.11 Resonence ............................................................................................. 37
27
Introduction
This is the second unit of physics-II for B.ed 2.5/4 years. This unit aims at explanation of
waves and phenomenon related to waves, group waves, sinusoidal waves, group speed
and dispersion, power and intensity in waves and related equations are described in a
very simple and understand way. The unit contains self assessment questions and all
necessary information regarding the wave and physical media.
Objectives
After completion of this unit you will be able to:
State the terms related to waves and physical media.
Differentiate between different types of waves
Elaborate the concept of phase velocity of travelling waves
Derive wave equation
Explain power and energy in wave motion
Identify and describe the principles of superposition
Give examples of resonance from real life.
28
2.1 Mechanical Waves
A mechanical wave is a wave that is an oscillation of matter, and therefore transfers
energy through a medium. While waves can move over long distances, the movement of
the medium of transmission, the material is limited. Therefore, the oscillating material
does not move far from its initial equilibrium position.
2.2 Types of Waves
There are three categories:
2.2.1 Longitudinal waves Type of waves, in which movement of the particles are parallel to the motion of the
energy. This type of waves are called longitudinal waves.
2.2.2 Transverse waves Type of waves, in which movement of the particles are at right angles (perpendicular) to
the motion of the energy. This is called transverse waves.
29
2.2.3 Surface waves It is the type of waves in which particles travel in a circular motion. This is called surface
waves.
2.3 Travelling Waves
A mechanical wave is produced by a vibrating body and travels through a medium from
one point to another, transporting energy as it moves. The mechanism by which a
mechanical wave propagates itself through a medium involves particle interaction; one
particle applies a push or pull on its adjacent neighbor, causing a displacement of that
neighbor from the equilibrium or rest position. As a wave is observed traveling through a
medium, a crest is seen moving along from particle to particle. This crest is followed by a
trough that is in turn followed by the next crest. In fact, one would observe a distinct
wave pattern (in the form of a sine wave) traveling through the medium. This sine wave
pattern continues to move in uninterrupted fashion until it encounters another wave along
the medium or until it encounters a boundary with another medium. This type of wave
pattern that is seen traveling through a medium is sometimes referred to as a traveling
wave.
30
2.4 Phase Velocity of Travelling Wave
The phase velocity of a wave is the rate at which the phase of the wave propagates in
space. This is the velocity at which the phase of any one frequency component of the
wave travels. For such a component, any given phase of the wave (for example, the crest)
will appear to travel at the phase velocity. The phase velocity is given in terms of the
wavelength λ (lambda) and period T as = Equivalently, in terms of the wave’s angular
frequency ω, which specifies angular change per unit of time, and wavenumber (or
angular wave number) k, which represents the proportionality between the angular
frequency ω and the linear speed (speed of propagation) νp,
2.5 Waves Sinusoidal Waves
2.5.1 Definition “Sinusoidal wave” is an adjective that means “a wave that resembles a sine graph.” In
synthesized sound a simple sine wave generator produces a "thin" sound that might be
useful in a device to test frequency response of somebody's ears, but can't compare with
even a cheap violin, much less a fine violin.
2.5.2 Mathematical Relation A sine wave or sinusoid is a mathematical curve that describes a smooth repetitive
oscillation. A sine wave is a continuous wave. It is named after the function sine, of
which it is the graph. It occurs often in pure and applied mathematics, as well as physics,
engineering, signal processing and many other fields. Its most basic form as a function of
time (t) is:
y(t) = Asin(2πft +φ ) = Asin(wt+φ)
where:
A = the amplitude, the peak deviation of the function from zero.
f = the ordinary frequency, the number of oscillations (cycles) that occur each second of
time.
ω = 2πf, the angular frequency, the rate of change of the function argument in units of
radians per second
31
φ = the phase, specifies (in radians) where in its cycle the oscillation is at t = 0.
When φ is non-zero, the entire waveform appears to be shifted in time by the amount φ
/ω seconds. A negative value represents a delay, and a positive value represents an
advance.
2.5.3 Graphical Representation
2.5.4 Physical Importance The sine wave is important in physics because it retains its wave shape when added to
another sine wave of the same frequency and arbitrary phase and magnitude. It is the only
periodic waveform that has this property. This property leads to its importance in Fourier
analysis and makes it acoustically unique
2.6 Group Speed and Dispersion
2.6.1 Group Speed The group speed of a wave is the speed with which the overall shape of the waves’
amplitudes—known as the modulation or envelope of the wave—propagates through
space.
For example, if a stone is thrown into the middle of a very still pond, a circular pattern of
waves with a quiescent center appears in the water, also known as a Capillary wave. The
expanding ring of waves is the wave group, within which one can discern individual
wavelets of differing wavelengths traveling at different speeds. The shorter waves travel
faster than the group as a whole, but their amplitudes diminish as they approach the
leading edge. The longer waves travel more slowly, and their amplitudes diminish as they
emerge from the trailing boundary of the group.
Frequency dispersion in groups of gravity waves on the surface of deep water. The red
dot moves with the phase velocity, and the green dots propagate with the group velocity.
32
In this deep-water case, the phase velocity is twice the group velocity. The red dot
overtakes two green dots when moving from the left to the right of the figure.
New waves seem to emerge at the back of a wave group, grow in amplitude until they are
at the center of the group, and vanish at the wave group front.
For surface gravity waves, the water particle velocities are much smaller than the phase
velocity, in most cases.
This shows a wave with the group velocity and phase velocity going in different
directions. The group velocity is positive (i.e. the envelope of the wave moves
rightward), while the phase velocity is negative (i.e. the peaks and troughs move
leftward).
2.6.2 Dispersion In fluid dynamics, dispersion of water waves generally refers to frequency dispersion,
which means that waves of different wavelengths travel at different phase speeds. Water
waves, in this context, are waves propagating on the water surface, with gravity and
surface tension as the restoring forces. As a result, water with a free surface is generally
considered to be a dispersive medium.
For a certain water depth, surface gravity waves – i.e. waves occurring at the air–water
interface and gravity as the only force restoring it to flatness – propagate faster with
increasing wavelength. On the other hand, for a given (fixed) wavelength, gravity waves
33
in deeper water have a larger phase speed than in shallower water. In contrast with the
behavior of gravity waves, capillary waves (i.e. only forced by surface tension) propagate
faster for shorter wavelengths.
Besides frequency dispersion, water waves also exhibit amplitude dispersion. This is a
nonlinear effect, by which waves of larger amplitude have a different phase speed from
small-amplitude waves.
2.7 Wave Equation
The wave equation is an important second-order linear hyperbolic partial differential
equation for the description of waves—as they occur in classical physics—such as sound
waves, light waves and water waves. It arises in fields like acoustics, electromagnetic,
and fluid dynamics.
Historically, the problem of a vibrating string such as that of a musical instrument was
studied by Jean le Rond d'Alembert, Leonhard Euler, Daniel Bernoulli, and Joseph-Louis
Lagrange. In1746, d’Alembert discovered the one-dimensional wave equation, and within
ten years Euler discovered the three-dimensional wave equation
The wave equation is a linear second-order partial differential equation which describes
the propagation of oscillations at a fixed speed in some quality u:
34
The equation is a good description for a wide range of phenomena because it is typically
used to model small oscillations about an equilibrium, for which systems can often be
well approximated by Hooke's law. Solutions to the wave equation are of course
important in fluid dynamics, but also play an important role in electromagnetism, optics,
gravitational physics, and heat transfer. Especially important are the solutions to the
Fourier transform of the wave equation, which define Fourier series, spherical harmonics,
and their generalizations.
2.8 Power and Intensity in Wave Motion
2.8.1 Power Power is the rate of transfer of energy by wave.
Brightness/loudness depends also on area receiving power:
Intensity, I = P/a= power-to-area ratio
2.8.2 Intensity The intensity (I) of any wave is the time-averaged power (P) it transfers per area (A)
through some region of space. The traditional way to indicate the time-averaged value of
a varying quantity is to enclose it in angle brackets ⟨⟩ . These look similar to the greater
and less than symbols but they are taller and less pointy. That gives us an equation that
looks like this…
I = ⟨P⟩/A
The SI unit of power is the watt, the SI unit of area is the square meter, so the SI unit of
intensity is the watt per square meter — a unit that has no special name.
The intensity of a sound wave is a combination of its rate and density of energy transfer.
It is an objective quantity associated with a wave. Loudness is a perceptual response to
the physical property of intensity. It is a subjective quality associated with a wave and is
a bit more complex. As a general rule the larger the amplitude, the greater the intensity,
the louder the sound. Sound waves with large amplitudes are said to be "loud". Sound
waves with small amplitudes are said to be "quiet" or "soft". (The word "low" is
sometimes also used to mean quiet, but this should be avoided. Use "low" to describe
sounds that are low in frequency.) Loudness will be discussed at the end of this section.
35
2.9 The Principle of Superposition
The Principle Of Superposition states that when two waves of the same kind meet at a
point in space, the resultant displacement at that point is the vector sum of the
displacements that the two waves would separately produce at that point.
2.10 Interference of Waves, Standing Waves
2.10.1 Interference When two or more waves simultaneously and independently travel through the same
medium at the same time, their effects are super positioned. The result of that
superposition is called interference. There are two types of interference: constructive and
destructive.
Constructive interference occurs when the wave amplitudes reinforce each other,
building a wave of even greater amplitude.
Destructive interference occurs when the wave amplitudes oppose each other, resulting
in waves of reduced amplitude.
36
2.10.2 Standing Waves When two identical waves travel through the same medium at the same time but in
opposite directions, a special interference pattern called a standing wave is formed.
Within a standing wave, regions of constructive interference are called antinodes and
regions of destructive interference are called nodes.
This name is derived from the impression that the wave appears to be "standing still"
since the nodes and antinodes are not being translated from one end of the medium to the
other even though the wave's energy is continuously traveling “back and forth.”
The lowest frequency to produce a standing wave pattern in a medium is called the
fundamental, or the 1st harmonic. As additional "loops" are inserted, overtones are
produced. A loop equals a distance of ½λ. In each case, since the medium has not
changed, the wave speed remains constant and we see evidence of the relationship that
the wavelength is inversely proportional to the frequency.
From the graph, For:
L = 1 loop
L = 0.5 λo
37
f = fo
L = 2 loops
L = 1.0 λ1
f1 = 2fo
L = 3 loops
L = 1.5 λ2
f2 = 3fo
2.11 Resonence
Resonance is a phenomenon in which a vibrating system or external force drives another
system to oscillate with greater amplitude at specific frequencies.
Frequencies at which the response amplitude is a relative maximum are known as the
system's resonant frequencies or resonance frequencies. At resonant frequencies, small
periodic driving forces have the ability to produce large amplitude oscillations, due to the
storage of vibrational energy.
38
Problems:
2.1 What happens to the power if the frequency is increased by a factor of 10? Ans
(100 )
2.2 Show that the grroup velocity of de borgile wave is given by Vg=Vp
=dE/dp?
2.3 Calculate de Broglie wavlength of an electron moving with
velocity 2×104m/s. [Given: mass of electron = 9.1 × 10-31
kg and Planck’s constant
= 6.6 × 10-34
Js]? Ans (3626A)
2.4 If the phase velocity of a ripple is by the relation vp = c1 + c2λ, where c1and c2 are
constant. Find the group velocity of the ripple? Ans (Vg=
Vp)
2.5 For a light of source is 60 watts, what is the intensity of light 10meters away from
source? Ans (.0478w/m)
2.6 If the distance from source doubles, how does intensity change? Ans (4)
39
UNIT-3
SPECIAL THEORY OF RELATIVITY
Written by: Hanifa Obaid
Reviewed by: Dr. Amjad Imran
40
CONTENTS
Introduction ....................................................................................................... 41
Objectives ......................................................................................................... 41
3.1 Inertia and Non Inertia .......................................................................... 42
3.2 Postulates of Relativity ......................................................................... 43
3.3 The Lorentz of Transformation of Derivative ...................................... 43
3.4 Assumptions on which Inverse Transformation Derived ..................... 45
3.5 Consequences of Lorentz Transformation ............................................ 45
3.6 Relativity of Length, Time and Mass ................................................... 46
3.7 Transformation of Velocity................................................................... 47
3.8 Variation of Mass with Velocity ........................................................... 48
3.9 Mass Energy Relation and its Importance ............................................ 49
3.10 Relativistic Momentum and Relativistic Energy .................................. 52
Problems ........................................................................................................... 55
References ......................................................................................................... 56
41
Introduction
Special theory of relativity is 3
rd unit of Physics-II for B.Ed 2.5/4 Year. In this unit
inertial and inertial frame is given. Einstein special theory of relativity is discussed in this
unit. The importance of Einstein in our life is described. Inertia, non-inertia and
relationship between mass and energy is elaborated in a very simple way. Relativity of
length, mass and time is described.
Objectives
After completion of this unit you will be able to:
1. Define the terms related to special theory of relativity.
2. Tell postulates of theory of relativity.
3. Derive the Lorentz Transformation; equation.
4. Explain the relation of mass and velocity.
5. Derive the relationship of mass, velocity and length.
6. Derive mass–energy relationship and factors related to it.
7. Describe relativistic momentum and Relativistic energy.
42
3.1 Inertia and Non Inertia
3.1.1 Inertia The scientist name Galileo give the basic concept of Inertia .In which he explained the
urgently stop of motion of moving body due to some force which is called friction. He
also performed his experiment with two inclined planes those place in the front of each
other and observed the rotation of ball along these planes.
Newton’s first law also favors his concept and states that to change the state of an object
need some force either its static or moving state.
The Inertia is defined as
“Inertia of body is its property due to which it resists any change in its state of rest or
motion.”
Mathematical Form:
I=mr I represent the moment of Inertia, m represent the mass and r for radius
3.1.1.2 Types of Inertia Inertia executes in many forms. Inertia can be described in context of direction , rest and
motion.
i. Inertia of Rest:
It is the inability of body to change its state of rest itself.
For example: A person will fall backward when the bus starts suddenly.
ii. Inertia of Motion:
It is the inability of body to change its state of motion itself.
For example: A passenger sitting in car will fall forward when the car suddenly
stops.
iii. Inertia of Direction:
It is the inability of the body to change its direction of motion itself.
For example: A person sitting inside the car will thrown outwards when car moves
in round curve.
43
3.1.2 Non Inertia When a body doesn’t seem to be accordance with inertia.
3.2 Postulates of Relativity
Albert Einstein Published “ZurElektrodynamikbewegterkorper in Annalen der physik
17(1905).this is the research paper in which Einstein established the Special theory of
Relativity.
There are two Postulates of relativity
1. The principle of relativity
The laws of physics are the same in all Inertial frame of reference.
2. The Constancy of speed of light in vacuum
The speed of light in vacuum has the same value of c in all Inertial frames of
reference.
C=3× m/s
The speed of light in Vacuum c (299792458 m/s ) is so enormous that we do not notice a
delay between transmission and reception of electromagnetic waves under normal
circumstances. The speed of light in vacuum is the actually only the speed that is absolute
and same for all observers
3.3 The Lorentz of Transformation of Derivative
In a 1964 paper ,[2]
Erik Christopher Zeema showed the causality preserving property ,a
condition is weaker mathematically than the invariance of the speed of light, is enough to
assure that the coordinates transformation is Lorentz transformation.
This transformation is defined as
“The coordinate transformations between two coordinate frames that move at constant
velocity relative to each other.”
The name of transformation is given after the Dutch physicist HedrickLorentz.
Suppose observers in frames S and S0 are moving with a velocity v relative to each other.
To be more precise, let's align the spatial coordinate systems for each observer, so S0 is
moving along the z-axis of observer S and the x and y axes are parallel with the x0 and
y0 axes. Suppose, also that both observers set their clocks so that when their origins are
on top of each other, their clocks read 0 also.
Then if an observer in frame S sees an event at the space time coordinate x, and an
observer S0 sees the same event at space-time coordinate x0 , the coordinates are related
by the Lorentz transformation.
44
We have
.
In term of t,x,y and z, the Lorentz transformation is
Ct’=
x’=x
y’=y
z’= (z-
As we mentioned, this transformation assures that if both observers see the same light ray
and measure its speed, they both get the same result, namely c. A simple way to check
this is to suppose that the light ray leaves the origin at time t = 0. Later on, the observer S
notices that it has reached the space-time coordinate x, while observer S0 notices that it
has reached x0 . The relationship between these observations is given by the Lorentz
transformation. It is easy to check the Lorentz invariance property.
1-(x
1)
2-(x
2)
3=(x
0t)
1-(x
1t)
2-(x
2t)
3 or
(ct)2- 2
=(ct’)2-(r’)
2.
where we have introduced the distance to the origin r = for observer S and
r’= for observer S’ . For a light ray moving with velocity c we must
have r = ct. So both sides of the equation above must be zero. More generally, Both
observers always get the same value for any Lorentz invariant quantity! So we conclude
that also r’= ct’ , and both observers agree that the speed is c.
If the relative speed v is much less than the speed of light, we have .
45
3.4 Assumptions on which Inverse Transform Derived
The Einstein established the special theory of relativity on two postulates and derived
many addition assumptions. These common assumptions are:
1. Assumptions of the invariance of relative velocity between two inertia coordinates.
If someone see other is moving away from him with constant velocity V , then first
one see other moving away from them.
2. Assumptions that lengths measured perpendicular to the relative motion of two
frames are not contracted; i.e x and z coordinates transform in a galilean manner.
3. Assumption that relations of the coordinate transformation are linear.
4. Assumption that space time is homogenous and isotropic.
3.5 Consequences of Lorentz Transformation
3.5.1 Relativity of Simultaneity
Suppose there are 2 types of reference frames S and S ‘ with S ‘ moving with velocity v
relative to S along positive X-axis.
Let the two events are said to be simultaneous if they occur at the same time. Let us
consider two events occur simultaneously in frames at P 1 and P 2.
Let ( , ) and ( ) be the coordinates of measurements made by an
observer at the origin of system S of the two events and respectively. Since, the
events are simultaneous in frame S, we have . If and are the corresponding
times of the same two events with respect to the system S ‘, then we have the following
relations from Lorentz transformations.
and =
---------------------------------------(1)
=
-
=
-
=
(2)
Then if events are simultaneous in frames S’, must be equal to . But is not so because
is not zero. So the same two events are not simultaneous in S’.
46
3.6 Relativity of Time, Length and Mass
3.6.1 Time Dilation (Einstein’s Time Dilation/Apparent Retardation of Clocks):
Consider there are two frames of reference S and S’, with S’ moving with relative
velocity v relative to S in the direction of X-axis and let this be placed at point
.
Let the clock be placed in reference system S which is at rest. Let this clock give a signal
at time in system S and suppose that is the time measured by the observer in
corresponding to
So the Lorentz transformation is
(3)
If the clock gives a signal is given at time is corresponding time in frame S’ is given
by
=
(4)
So, the clock gives a signal at an interval = in system S. when the signal is
measured from the moving S’ is equal to = .
So the equations (3) and (4) we have
=
= (
)
) (5)
It is clear from equation 5 that time interval appears to the moving observer to be dilated
by a factor
.
47
3.6.2 Relativity of Space/Length Contraction Assume that coordinates of end of road are , . Let = in moving S’ and
L= .
Then the equation is
=(
) and ==(
) + vt
From the above equation
=(
) - )
Or L = (
) , L<
3.6.3 Relativity of Mass The increase in effective mass with speed is given by expression
m=
=
it follows the Lorentz transformation when collide are describe from a fixed and moving
reference frame , where it arises as a result of conservation of momentum. You can also
approach it from the relativistic energy expression
E=mc2
Where the effective mass is given by energy divided by c2. This suggests that this “m” is
really about the energy, and that the real mass is the rest mass
3.7 Transformation of Velocity
No two object have relative velocity greater than c. if we observer a spacecraft travelling
at 0.9c and it fires a projectile which it observes to be moving at 0.6c with respect to it?
The velocities must be transform into the Lorentz transformation and that leads to be a
very non-intuitive result called Einstein velocity addition.
Assume an object moving with velocity u with respect to inertia frame S and S’ moving
with velocity v respect to S.
Then U’=
(6)
48
The Lorentz transformation is
x’= (x-vt)-------------------------------------------------------- (7)
t’= (t-
)---------------------------------------------------------(8)
where
just taking a differentials of these two equation leads to the velocity transformation.
Taking the differentials of Lorentz transformation expression for x’ and t’ .
=
Putting this in the notation introduced in the above equation (6)
U’=
The reverse transformation is obtained by solving U in above expression.
U’=
.
Applying this transformation in spacecraft travelling at 0.9c we obtained a Velocity.
3.8 Variation of Mass with Velocity
According to Newtonian mechanics the mass of a body does not change with velocity.
However, conservation laws, especially here the law of conservation of momentum, hold
for any inertial system. Hence, in order to maintain the momentum conserved in any
isolated system, mass of the body must be related to its velocity. So according to
Einstein, the mass of the body in motion is different from the mass of the body at rest.
We consider two inertial frames S and S.
We now consider the collision of two bodies in S’ and view it from the S. Let the two
particles of masses m1 and m2 are travelling with velocity u ‘ and-u ‘ parallel to x-axis in
S’. The two bodies collide and after collision they coalesced into one body.
In System S: Before Collision: Mass of bodies are m1 and m2• Let the their velocities
are u1 and u2 respectively
In System S: After Collision: Mass of the coalesced body is (m1+ m2) and the velocity Is
v.
49
Using law of addition vector
U=
----------(9) and
U’=
-----------(10)
Applying the principle of conservation of momentum of the system before and after the
collision, we have,
m1 u1 +m2 u2 = (m1 +m2)v
m1 =[
]+ m2 [
] =( m1 + m2 )v
=[
]
now using equation we have
m1/m2=[
/
-----------------------------------------(11)
Let the body of mass m2 is moving with zero velocity in S before collision, i.e., u2 = 0
Then equation 11 become
m1/m2=
using common notation m1=m , m2=mo and u=v
m=mo/ .
This is the formula of variation of mass with velocity. mois the rest mass of body and m is
the relativistic mass of body.
3.9 Mass Energy Relation and its Importance
Einstein correctly described the equivalence of mass and energy as “the most important
upshot of the special theory of relativity” (Einstein, 1919), for this result lies at the core
of modern physics. According to Einstein's famous equation E = mc2, the energy E of a
physical system is numerically equal to the product of its mass m and the speed of
light c squared. It is customary to refer to this result as “the equivalence of mass and
energy,” or simply “mass-energy equivalence,” because one can choose units in
which c = 1, and hence E = m.
50
The two main philosophical questions surrounding Einstein's equation concern how we
ought to understand the assertion that mass and energy are in some sense equivalent and
how we ought to understand assertions concerning the convertibility of mass into energy.
In this entry, we discuss the physics E = mc2 and its application.
3.9.1 The Physics of E = mc2:
We first present a minimal interpretation of E = mc2. The interpretation is minimal in the
sense that it makes as few metaphysical and ontological commitments as possible.
Furthermore, it is an interpretation with which nearly all physicists and philosophers now
agree. We then illustrate the physical implications of Einstein's equation by considering
three typical examples of mass-energy equivalence at work.
3.9.2 A minimal interpretation of E = mc2:
To interpret E = mc2 we first need to understand the meaning of the symbols E and m.
Unfortunately, these symbols are not used univocally by physicists and philosophers.
However, a common interpretation, which we shall adopt for now, is that E represents the
total energy of a physical system S. The symbol m represents the relativistic mass of S,
which is the mass of S as measured by an observer O that moves with a constant
velocity v relative to S.
In the special case where O and S are in a state of relative rest the mass of S measured
by O is called the rest-mass, which one often designates mo. The rest-mass of S is a
measure of the inertia of S, i.e., of the tendency of S to resist changes in velocity. Thus,
the rest-mass of S is simply its inertial mass. The rest-mass is related to the relativistic
mass by the equation:
m = mo γ(v) ----------------------------------------------(12)
The value of the energy we obtain from Einstein's equation when S and O are in a state of
relative rest is often called the rest-energy and is commonly designated Eo. Significantly,
Einstein often called the rest-energy the “energy content [Energieinhalt]” (1905b) of the
body, as it is a measure of all of the energy, including the potential energy, of the
constituents of S.
51
We can display the relationships between the various masses and energies we have
discussed by writing down Einstein's equation in the following form:
E = mo γ(v) c2 -------------------------------------------------(13)
where we have simply substituted the expression for the relativistic mass from (12)
into E = mc2. In the rest frame of S the Lorentz factor is 1, because the
velocity v of S relative to its own rest frame is zero. Consequently, when O and S are in a
state of relative rest equation 13 ,become
Eo = mo c2-----------------------------------------------------(14)
where we write Eo on the left-side of (3) to indicate that we are referring to the rest-
energy of S.
We shall henceforth be referring exclusively to the result expressed by (14).
Consequently, we shall simply call Eo “the energy of S” and mo “the mass of S” unless we
need to qualify these expressions either to avoid ambiguity or to emphasize a particular
feature of some result. Furthermore, we shall follow the fairly common practice in the
physics literature of dropping the subscript “o” from Eo and mo respectively. Thus, from
now on, we will use E to designate rest-energy and m to designate rest-mass.
The result Einstein originally derived in (1905b) is sometimes called (e.g., by Baierlein,
2007) the “incremental” version of (14), which we can now write using our recently
adopted conventions for the symbols E and m as:
ΔE = Δm c2------------------------------------------------------------------(15)
where ΔE designates a change in the energy of S and Δm designates a change in the mass
of S. If we re-write (16) by dividing both sides by c2, we can see that (16) also says that if
the energy of S changes by an amount ΔE, its mass changes concurrently by an amount
Δm = ΔE/c2.
3.9.2 Importance or Application of ΔE = Δm c2
To illustrate the physical implications of (15), physicists tend to use two main types of
examples:
(i) examples that examine the mass and energy of a single body as an un-analyzed
whole,
(ii) examples that examine the mass and energy of a collection of objects, especially,
atomic and sub-atomic objects involved in collisions. The latter class of examples is
particularly useful for understanding mass-energy equivalence because such examples
deal with changes in energy es and masses that are detectable. Furthermore, such
examples can emphasize the importance of considering all the energy of a physical
system, including the potential energy of its constituents, in calculating the total mass
(and energy) of that system.
(iii) Its importance is in Mass and energy of an ideal gas.
52
3.10 Relativistic momentum and Relativistic energy
3.10.1 Relativistic Momentum Newton’s 2nd Law can be written in the form
F =dp/ dt
where the non-relativistic momentum of a body is p = mu where u = dx/ dt .
However, because of the Lorentz transformation equations, dx/ dt is measured
differently in different inertial frames. Thus, Newton’s 2nd Law would not have the same
form in different frames. We need a new definition of momentum to retain the definition
of force as a change in momentum. Suppose p = m dx/ dτ , where τ is the proper time in
the object’s rest frame. Every observer will agree on which frame is the rest frame. Also,
since y =y ′ and z′ = z, the transverse momentum (py and pz) will be invariant for a
Lorentz transformation along the x axis. (This would not be the case if we did not use the
proper time in the definition). We can rewrite this momentum definition as follows
P=m
=m
t=
where
from the time dilation
p= mu
=
.
Note that u is the velocity of the object in a reference frame, not the velocity of a
reference frame relative to another.
Note that u is the velocity of the object in a reference frame, not the velocity of a
reference frame relative to another. In this definition of momentum, the mass m=m0 is
the “rest mass”. That is, it is the mass of an object in its rest frame. Sometimes γ m is
referred to as the “relativistic mass”, such that we can retain the Newtonian definition of
momentum as p = mu. In this sense, the mass of an object grows as its velocity increases.
But this convenient trick can be problematic. As we shall see, the kinetic energy, for
example, is not ½ m .
53
3.10.2 Relativistic Energy Now work is defined as force applied over a distance. It corresponds to the expended
energy to accelerate a body. If the force and path are constant,
W =F⋅d
More generally, if the force and path vary, then a line integral must be performed from
initial position 1 to final position 2.
W12=
The work applied to a body translates to a change in the kinetic energy since energy must
be conserved. If we assume that the body is initially at rest, then the final kinetic energy
is equal to the work expended:
W= K=
( mu).udt where ds=u dt
W= K=
( u).u
W= K=
Integrate by parts
W= K=
W= K=
= +m u0
After solve we get
W=K= m
W=K=
This final expression for the kinetic energy looks like nothing like the non-relativistic
equation K=1/2m . However, if we consider velocities much less than the speed of
light, we can see the correspondence.
So at low velocities there is no difference between the definition of kinetic energy in
Special Relativity from that in Newtonian Mechanics. Now let’s consider the opposite
limit when the velocity approaches the speed of light. In that case, the kinetic energy
becomes infinite as the relativistic factor γ goes to infinity. This is another way of saying
that objects cannot exceed the speed of light, because it would take an infinite amount of
energy. Now let’s rewrite the equation involving the kinetic energy.
54
E=
This equation has the form of kinetic energy plus potential energy equals total energy.
What is the potential energy? It is the term.
E0= .
which we refer to as the rest energy. As you know, this is Einstein’s famous equation that
tells us that mass is another form of energy. Mass can be converted into energy and vice
versa. How much energy? Let’s see
Example: Suppose that a 1 kg mass moves at a velocity u = 1 m/s. The kinetic energy is
½ m u 2= ½ J. (We can use the non-relativistic equation because the velocity is much
much smaller than the speed of light.) The rest mass energy is =9.0 . Clearly
there is a tremendous amount of energy in 1 kg of mass. That is why nuclear weapons
have the power that they do, because they convert a significant amount of mass into
energy.
55
Problems 3.1 In an inertial frame S, two laser pulses are emitted by points on the x-axis 10 km
apart and separated by 3µs. They reach an inertial observer O travelling in standard
configuration with velocity v away from S. O receives the two laser pulses
simultaneously. Find v.
3.2 Write the mathematical expression of a Lorentz boost with the{ ct , x , y ,z }inertial
frame sliding along the z- (or z -) axis, and with the x - and the y - axes parallel to
the x-axis and the y-axis, respectively.
3.3 Show that the rapidity φ satisfies the relations
eφ = γ (1 + β),
e−φ
= γ (1 − β),
γ = cosh φ, βγ = sinh φ,
so that the Lorentz transformation can be written as
ct = ctcosh φ − x sinh φ,
x = −ctsinh φ + x cosh φ.
56
References 1. Physics 5110 Lorentz Transformations {March 14, 2010}.
2. College of physics science and technology, Heilongjiang University, Harbin
150080, People’s Republic of china.
57
UNIT-4
LIGHT
Written by: Dr. Amjad Imran
Reviewed by: Hanifa Ubaid
58
CONTENTS
Introduction ....................................................................................................... 59
Objectives ......................................................................................................... 59
4.1 Nature of Light ...................................................................................... 60
4.2 Light as an Electromagnetic Wave ....................................................... 61
4.3 Characteristics of Light Waves ............................................................. 62
4.4 Electromagnetic Spectrum .................................................................... 64
4.5 Speed of Light in Matter ....................................................................... 65
4.6 Answers to Self-Assessment Questions ................................................ 66
References ......................................................................................................... 66
59
Introduction
What is light? This question has been much talked-about for many centuries. The sun
emits light that brightens our darkness. The answer, in short, is that the light is a special
kind of energy. The speed of light, although very fast, but is not infinite. It is
m/s in vacuum. One of its well-known characteristics is that it travels in vacuum in a
straight line at a constant speed thus carrying energy from one point to another.
Light plays an important role in our daily lives in this era of technology. It is used in
compact disc (CD) players, in which a laser (a special king of light) reflecting off a CD
transforms the returning signal into music. It is used in cash and carries (Shopping
Stores), where it is used to read bar codes for prices. It is also used in laser (a kind of
light) printers to record images on paper. It is used in digital cameras that capture our
world. It is used in fiber-optic cables that connect computers and telephones to one
another. And light is used in the treatment of the diseases, to produce images used in
hospitals.
Objectives
After studying this unit, students will be able to
1. Describe the nature of light
2. Explain different theories of light like corpuscular theory, Huygens theory and their
limitations and drawbacks
3. Describe the Maxwell’s contributions in explaining nature of Light
4. Understand different characteristics of Light like Amplitude of a wave, Wave
Number, Frequency of a wave and Time period
5. Explain different regions of Electromagnetic spectrum, for instance, Microwave
Region, Infrared Region, Visible region, Ultraviolet Region, X-Rays, Gama Rays
6. Understand on which factors Speed of Light depends upon?
7. Illustrate the relationship between Speed of Light and refractive index of the matter
60
4.1 Nature of Light
Until the beginning of 19th century, light was considered as a stream of particles emitted
by the light source. According to this model the light particles stimulates the sense of
sight on entering the eye. This model was presented by the great scientist Isaac Newton.
This model provided some justifications to some facts that were known at that time,
regarding the nature of light namely i.e the laws of reflection and refraction.
In spite of the virtues of the particle theory of light, thinkers like Huygens realized that
there are some other experimental facts that could not be explained on the basis of this
theory. For example
1. It was unable to explain partial reflection and refraction from the surface of a
transparent medium (as a particle can either reflect or transmit through an interface
but cannot exhibit both properties simultaneously).
2. It could not explain phenomenon such as interference, diffraction, polarization etc.
3. The particle theory predicted that speed of light in a denser medium is greater than
the speed of light in a rarer medium that was experimentally proved wrong by
Foucault.
4. If one makes a pinhole very small, the image of the light going through begins to
widen, instead of narrowing.
Newton’s attempts to explain simultaneous reflection and refraction from a transparent
surface were not considered satisfactory by a large group of thinkers.
During the Newton’s life time, however, another model was suggested – a model that
views light as having wave-like properties. In 1678, Dutch physicist and astronomer
christiaan Huygens proved that a wave model of light can also explain the laws of
reflection and refraction. The wave model did not received acceptance for several years.
All the waves known at that time (sound, water etc) need medium to travel but light from
sun could travel to earth through the space that was considered empty. Even though wave
nature of light was proved by Grimaldi (1618-1663) around 1660, most scientists rejected
the wave model for more than a century and adhered to Newton’s particle model.
The first clear and convincible demonstration of wave nature of light was provided in
1801 by Thomas Young (1773-1829) who showed that under appropriate conditions light
exhibits interference. That is, light wave emitted by a single source and travelling along
two different paths can arrive some point, combine and cancel each other by destructive
interference. Such behavior could not be explained at that time by particle model as
scientist could not imagine how two or more particles could come together and cancel
one another. Additional developments during the 19th century led to the general
acceptance of wave model of light.
Although wave model is assumed to be well established and could explain most known
properties of light but some experiments could not be explained by the assumption of wave
61
nature of light. The most important was the Photoelectric effect that was discovered by Hertz,
in which electrons are emitted from the metal when its surface is exposed to light.
In view of the above discussion, light must be regarded as dual nature. In some
experiments it behave like a wave, and in others, it acts like particle
Self Assessment Questions 4.1 Q.1 What are the salient features of Newton’s Corpuscular Theory?
Q.2 Explain the drawbacks of the Corpuscular Theory
Q.3 Illustrate the Huygens Principles of Light
Q.4 Describe the Thomas Young’s’ contributions in explaining the wave nature of light
Q.5 Why wave nature of light failed to explain Photoelectric Effect?
4.2 Light as an Electromagnetic Wave
It has been pointed out in the last section that light has wave-like properties. But if light is
a wave, what is the medium through which light waves propagate even in outer space?
What is the nature of the disturbance that is propagated? What provides the restoring
force that causes the propagation? These questions were answered in 1864 by the most
brilliant and profound thinker of nineteenth century physics, James Clerk Maxwell.
Maxwell mathematically proved that light is a form of high frequency electromagnetic
wave. The electromagnetic waves consist of oscillating electric and magnetic fields. The
changing electric field produces varying magnetic field (Faraday’s Law) and varying
magnetic field, in turn, produces varying electric field (Ampere’s Law) which maintains
the propagation of the wave. Therefore, medium in the case of electromagnetic wave is
fields itself-that is why it can propagate through vacuum (requires no material medium)
unlike sound waves. He also predicted that electric and magnetic fields in
electromagnetic waves are at right angle to each other and to the direction of propagation
of wave and that these time dependent fields satisfy linear wave equation. He further
predicted that the velocity of light in medium
depends upon the constitutive
parameters of the medium i.e permittivity and permeability of the medium. From this
expression he calculated the speed of light in vacuum as m/s.
Figure 4.1: Direction of Electric and Magnetic Fields in Electromagnetic Wave
62
4.3 Characteristics of Light Waves
As described in the proceeding section, an electromagnetic wave is constituted of an
electric field and a magnetic field. The directions of the fields are at right angles to the
direction the wave is moving, just as water waves in which water moves up and down
while the water wave moves horizontally (i.e at right angle). The figure below shows
one-dimensional representation of the electric field
Figure 4.2: Wave form
The maximum value of the wave displacement along the vertical is called the Amplitude
of the wave. Thewave starts at zero and repeats after a certain distance. This distance is
called the Wave length . Lightcan have different wavelengths such as, the blue light has
smaller wave length as compared to blue light as shown in figure above. The inverse of
the wavelength is the Wave Number which is expressed in . The
wavepropagates at a speed called the Wave Speed . This wave speed in a vacuum is
equal to c, and is less than c in a medium. The wave passes by a stationary point in a
repeating cycle. The time to complete one cycle is called the Cycle Time or Period (τ)
and can be calculated using the expression
. Another important measure of a wave
is its frequency (f). It is measured as the number of waves that pass a given point in one
second. The unit for frequency is cycles per second, also called hertz (Hz). It can be
noticed that the frequency and the period are reciprocals of one another. If the wave
speed and wavelength are known, the frequency can be calculated using the following
expression
63
Self Assessment Questions Q.1 Circular water waves such as those shown in Figure below move outward from an
oscillatingcork at A. The cork moves up and down and takes one second to
complete an oscillatory motion, and generates waves that measure 10 cm from crest
to crest. Sometimes after the wave motion has been established, we begin to time
the motion with a stopwatch. At a certain time t = 10 s on the watch, we notice that
the wave profile has the shape shown below.
a. What is the wave frequency for this water wave?
b. What is its wavelength λ?
c. What is its wave speed ?
d. What is the phase angle for a wave front at position r = 102.5 cm at time t =
10 s?
e. What is the wave displacement y on the wave front at r = 102.5 cm?
f. What is the phase angle for a wave front at r = 107.5 cm at t = 10 s?
Q.2 A wave has speed 50m/s and wavelength 2m. Calculate its period
Q.3 Calculate the minimum distance between two points on the wave that differ in
phase by 60 degree when it is travelling through air.
Q.4 If wavelength of a wave in water is 0.05m, then calculate the speed of the wave in
water.
Q.5 What is the frequency of a wave that has speed of 0.4 m/s and the wavelength
0.020 m?
Q.6 Which of the following terms describes the number of waves that can be produced
over a given time?
A. Wavelength B. Pitch C. Frequency D. Amplitude
64
4.4 Electromagnetic Spectrum
When white light is passed through a prism, the emerging light is separated into colors
that form a spectrum. Newton was the first scientist, who showed that prism does not add
colors to the light, as we previously thought. These colors are already present in the white
light and the prism make them separated. He showed this by using the second prism to
recombine the colors produced by the first.
Electromagnetic waves have wide range in wavelength from very long to very short. This
entire range is called the Electromagnetic Spectrum. This spectrum has been divided into
many regions for our convenience on the basis of wavelength and energy. Rotating generators
and power lines produce low-frequency waves. The wavelengths of these waves are of the
order of 105 to 108 meters. Then come the radio waves that ranges from 0.3 to 105 meters.
Radio waves are used in commercial and radar communications. The microwave region
ranges from 0.01 to 0.3 meter and are used in radar and satellite communications and
microwave ovens. The infrared region, from 1 µm to 30 µm, was first detected by Sir
William Herschel in 1800. This region is subdivided into five regions: very near (1–3 µm),
near (3–5 µm), mid (5–6 µm), far (6–15 µm), and very long (15–30 µm) infrared. Just as the
ear cannot hear outside a small range of frequencies, similarly human eye cannot detect light
outside a small range of wavelengths (0.76–0.49 µm). This region is called the visible region.
The ultraviolet region (ranging from 400nm to 10nm) is a higher-energy region discovered
by Johann Ritter. It stimulates many chemical reactions and produces ionization in the upper
atmosphere thus creating the ionosphere. Being high energy waves, they can penetrate
through living cell and cause them to damage. They are also used in sterilization as they can
kill bacteria. X-ray regime was discovered by Wilhelm Röntgen in 1895. The wavelength
ranges from to meters. It has very high penetration power and passes through
flesh. It therefore produces an image of higher-density material such as bones. Gamma rays
have the smallest wavelength (less than meter). They showparticle-like properties.
These are emitted by the sun, linear and particle beam accelerators, and nuclear processes.
Figure 4.3: Electromagnetic Spectrum
65
Self Assessment Questions Q.1 What is Electromagnetic Spectrum?
Q.2 What type of electromagnetic waves has Longest and Shortest wavelength
Q.3 Explain some of the uses of Microwaves, ultrawaves, infrared rays, X-rays and
Gama-Rays
Q.4 What variables affect the speed of light?
Q.5 What are the harmful effects of excessive exposure to ultra waves, infrared rays, X-
rays and Gama-Rays
4.5 Speed of Light in Matter
The speed of light in vacuum, commonly denoted c, is the speed at which light passes
through space. Its exact value is 29,97,92,458m/s (approximately m/s or
186,000 mi/s). It is also the speed at which all mass less particles (photons and gluons
etc) and electromagnetic as well as gravitational waves pass through space.
The speed of light in the material medium such as glass or air, is less than c. Similarly the
speed of electromagnetic waves in wire cables is also slower than c. The parameter that
refers to the speed of light in a medium is the refractive index of the medium. It is the
ratio between c and the speed at which lighttravels in a material and is designated by
such that
. For example, for visible light the refractiveindex of glass isaround 1.5. It
means that light in glass travels at
. Similarly the refractive
index of air for visible light is about 1.0003. Therefore the speed of light in air is about
299,700 km/s.
It seems that light and other electromagnetic waves appear to reach its destination
instantaneously, but sensitive measurements reveals that it is not the case. There is a
substantial time delay for long distances. In communicating with distant space probes, it
can take minutes to hours for a message to get from Earth to the spacecraft, or vice versa.
The light that reaches on earth from distant stars left them many centuries ago. The finite
speed of light also limits the theoretical maximum speed of computers.
In 1865,James Clerk Maxwell suggested that light is an electromagnetic wave and it
travels through the space at speed
where are the constitutive parameters
of free space.
Ole Rømer was the scientist who first demonstrated that light travels at a finite speed (as
opposed to instantaneously) by studying the motion of Jupiter's moon Io.
Self Assessment Questions Q.1 On which factors speed of light depends upon?
Q.2 The refractive index of glass is 1.50. Calculate the speed of light in glass
66
Q.3 The refractive index of water is 1.33. The speed of light in air is m/s.
Calculate the speed of light in water.
Q.4 In which of these situations would light most likely slow down?
A. From glass to air B. From ice to water C. From plastic to air D. From a vacuum
to water
Q.5 Light travels fastest in
A. Warm air. B. Cool air. C. A vacuum.
4.6 Answers to Self-Assessment Exercises
Answers to Self-Assessment Exercise 4.3 Q.1
a. cycle/s,
b. 10cm,
c. 10cm/s,
d.
,
e. crest,
f.
Q.2 0.04s,
Q.3 0.008m,
Q.4 1500m/s
Q.6 C
Answers to Self-Assessment Exercise No. 4.5 Q.2 m/s
Q.3 m/s
Q.4 D
Q.5 C
References
1. Halliday, Resnik, and Krane, “ Fundamentals of Physics”, Wiley and Sons, Inc
2. Serway and Jewett. “Physics- A Calculus-Based Approach”, Books Cole in.
3. Sears, Zemansky and Young, “College Physics”, Amazon Publishers
4. Nelkon and Parker, “Advance Level Physics”, Hienemann Educational Books,
London
67
UNIT-5
INTERFERENCE
Written by: Dr. Amjad Imran
Reviewed by: Imran Amin Asim
68
CONTENTS
Introduction ....................................................................................................... 69
Objectives ......................................................................................................... 69
5.1 Interference of Light Waves ................................................................. 70
5.2 Coherent Sources .................................................................................. 70
5.3 Double Slit Experiment......................................................................... 71
5.4 Interference in Thin Films .................................................................... 73
5.5 Newton’s Rings ..................................................................................... 75
5.6 Answers to Self Assessment Questions ................................................ 76
References ......................................................................................................... 76
69
Introduction
In this chapter we’ll study that, when waves from two or more sources reach at a given
point in space, what happens to them. What is the constructive and destructive
interference? How Young and Newton demonstrated these phenomena?
If the two waves, reach at the given point in space then the total wave, is the sum of the
waves. The waves add, constructively or destructively, depend upon their phase
difference. If the waves are in phase, their amplitudes add up and hence constructive
interference occurs but if they are out of phase, their amplitudes cancel each other and
destructive interference takes place In the general case of many waves, we add them all
keeping in view of their phases.
In this chapter we will also discuss what the bright and dark fringes are and how they are
produced? On which factors they depend upon? How we calculate the path difference
and phase difference? The results in this chapter may be applied to any kind of wave (like
sound waves), but we will work in terms of light waves or generally, electromagnetic
waves.
Objectives
After studying this chapter, students will be able to:
1. Understand the interference of light
2. Distinguish between Constructive Interference and Destructive Interference
3. Illustrate the concept of coherent sources and how these sources can be realized
4. Describe that how bright and dark fringes are formed in Double Slit Experiment,
a. What are the Conditions of Constructive and Destructive Interference
b. That how the position of Dark and Bright Fringes can be determined on the
screen
5. Define the Interference in thin Films
a. The concept of Path Difference and Phase Difference
b. Conditions of Constructive and Destructive Interference
6. Explain the Newton’s Rings and how they are formed?
70
5.1 Interference of Light Waves
As discussed in the last chapter that light is the form of energy which is distributed
uniformly in the surrounding medium. If two sources of light giving out continuous
waves of same amplitude are held close to each other, then the distribution of energy is
not uniform in the surrounding. At some places the two waves reinforce and at some
other points they cancel each other’s effect. This non-uniform distribution of light energy
due to superposition of two or more waves is called as interference.
If the two waves are combined in such a way that the amplitude of the resultant wave is
greater than that of either of the individual wave, this kind of interference is called
constructive interference. Contrary to this if two waves combine so as to produce a
resultant wave of amplitude less than the amplitude of the individual waves, the
interference is called destructive interference.
Activity: Perform an activity to find the constructive interference and destructive interference.
Self Assessment Questions Q.1 Define constructive and destructive interference
Q.2 Give some applications of both types of interference
Q.3 What phenomenon is used in the automobile silencers?
5.2 Coherent Sources
Today we can produce interference effects very easily. In the days of Sir Isaac Newton
and Christian Huygens, however, demonstration of light interference was not an easy
task. There were several reasons for this. One of the basic reasons was that visible light
has short wavelength and to observe and detect interference patters of such a shorter
wavelength waves was not an easy job. Another reason was based on the difficulty of
generating coherent waves, that is, waves with a phase relationship with each other that
do not change during the time when interference was observed.
There are generally, two ways to produce coherent sources
Develop several coherent virtual sources from a single incoherent “point” source
with the help of mirrors then allow light from the two virtual sources to overlap and
interfere. (This method is used, for example, in the Loyd’s mirror experiment.)
Take monochromatic light from a single “point” source and pass it through two
small openings or slits and then allow light from the two slits to overlap on a screen
and interfere
71
5.3 Double Slit Experiment
Figure below demonstrates the interference with coherent light from two slits S1 and S2.
The source that is designated by, S0 is a monochromatic point source of light (source
producing light of single frequency) whose spherical wave fronts fall on the two slits to
create secondary sources S1 and S2. The sources S1 and S2 behaves as coherent sources so
that spherical waves emitting from the two sources maintain a fixed phase relationship
with each other as they spread out and fall on the screen, to generate a series of alternate
bright and dark regions. These alternate regions of bright and dark are termed as
interference fringes.
Figure 5.1: Young Double Slit Experiment
We can calculate the positions of the alternative bright and dark regions as shown in the
above figure. To do this we shall make use of the following figure
Figure 5.2: Geometry of the Problem
Light waves from slit S1 and S2 arrive at an arbitrary point P on the screen. If the two
waves are in phase, there will be constructive interference and we will observe a bright
spot at P; if they are out of phase, there will be destructive interference and we will
observe a dark spot. So the phase difference between the two waves arriving at point P is
a very important factor that will determine what will happen at point P. Path difference
between two paths S2P and S1P may be written as
72
If the path difference is equal to λ (wave length) or some integral multiple of λ, the two
waves arrive at P in phase and a bright fringe (bright spot) appears there. The condition
for bright fringes is, then,
(5.3.1)
where The number m is called the order number. The central bright
fringe at θ = 0 (point 0′ on the screen) is called the zeroth-order maximum . The
fringes corresponding to , are called the first-order maxima, and so on. If, on
the other hand, the path difference at P is an odd multiple of λ/2, the two waves arrive out
of phase and create a dark fringe or spot. The condition for dark fringes is given by
(5.3.2)
where
5.3.1 Position of Bright and Dark Fringes on the Screen
From the triangle
If is very small then, . If we put the value of in the above two
expressions we can get the expression for the position of bright and dark fringes
on the screen as follow
(5.3.4)
(5.3.5)
The distance between two consecutive bright or dark fringes is called the fringe spacing.
From the above two expressions it is clear that the fringe spacing is
. This expression may
be used to find out the wave length of the incident light in the double slit experiment.
Self-Assessment Questions Q.1 In the double-slit experiment, the two coherent sources with slit separation mm
are located m from a screen. The distance between successive bright fringes on
the screen is measured to be mm. What is the wavelength of the light?
Q.2 Suppose in the double-slit arrangement, mm, cm,
nm, and
(a) Determine the path difference for the rays from the two slits?
(b) Find out this path difference in terms of
73
Q.3 Two slits in double slit experiment have width in ratio . What will be the ratio
of the amplitudes?
Q.4 In Double-Slit experiment,
(a) Why do we use monochromatic light?
(b) If white light is used, how would the pattern change?
Q.5 In a Double-slit experiment, the slits are mm apart. Fringes in sodium are
observed on a screen that is placed at a distance of 1.2 m from the slits. If the
distance between the fringes is 2.4 mm, determine the wavelength of sodium light?
Q.6 In the above experiment, the same light gives a fringe spacing of 2 mm when
passed through a different pair of slits. Determine the slit separation if the distance
between the slits and the fringes is the same?
Q.7 In the experiment, the slits are 0.3 mm apart. White light passes through the slits and
fringes are observed at a distance of 2 m from the slits. Red light has a of 700 nm
and blue light has a of 400 nm. Calculate the fringe spacing for each color.
5.4 Interference in Thin Films
Rainbow of colors that we often see on thin-film soap bubbles and “oil slicks” is the result of
interference of white light. These are produced when the light reflecting from opposite
surfaces of the thin film, interfere. Thin films have great utility in optics. When thin films of
different refractive indexes and thicknesses are judiciously stacked, coatings can be created
that either enhance reflection greatly (HR coats) or suppress reflection (AR coats). To make
our discussion simple, we consider interference from a single thin film.
5.4.1 Single-film interference The geometry for thin-film interference is shown in figure below. We suppose that the
light is normally incident on a film of thickness and refractive index . While
discussing the interference, we will made use of the following established facts:
A light wave travelling from a medium of lower refractive index to a medium of
higher refractive index automatically undergoes a phase change of π (180°) upon
reflection. A light wave travelling from a medium of higher index to one of lower
index undergoes no phase change upon reflection.
The wavelength of light in a medium of refractive index is given by =
, where is the wavelength in a vacuum or, approximately, in air
Figure 5.3: Ray diagram
74
In figure above, it can be seen that a light ray in medium of index falls on a
transparent film of index that is placed on a substrate of index . For the simplicity,
we take the initial medium as air, so that . The light ray, which incidents on the
film surface at A, divides into two parts; a reflected and refracted rays. The refracted
ray reflects again into the thin film at the film-substrate interface at B and leaves the
film at C.
Since the two rays travel different paths from point A therefore, they develop path as well
as phase difference that contribute to their condition of constructive or destructive
interference.
Path Difference: The beam that reflects from point B travels more as compare to the
reflected beam. So the relative path difference, develops between the two beams
(5.4.1)
If the incident light normally strikes the surface of thin film, then
(5.4.2)
Phase Difference: Suppose that and . Often, in practice, , i.e
upper and lower media are the same, as in the case of a water film (soap bubble) in air.
As the light ray falls on the thin film, it reflects two times. A reflection occurs at point A
as the light goes from a medium of lower index to the medium of higher index .
And the second reflection occurs, at the lower surface of the film, when light moves from
the medium of higher index to the medium of lower index . Thus, the light
reflecting from the upper surface of the film, undergoes a shifts phase by 180°
(equivalent to one-half wavelength) while the light reflecting from the lower surface,
does not.
Therefor if as the equivalent phase difference introduced upon reflection, the
condition for constructive interference becomes
(5.4.3)
Or (for normal incidence)
(5.4.4)
And for destructive interference
(5.4.5)
where and is the wavelength of monochromatic light in air.
75
Self Assessment Questions Q.1 White light is incident normally on the surface of a soap bubble. A portion of the
surface reflects green light of wavelength = 540 nm. Assume that the refractive
index of the soap film is near that of water, so that = 1.33. Estimate the
thickness (in nanometers) of the soap bubble surface that appears green in second
order.
Q.2 A thin piece of glass with an index of refraction of is placed on top of a
medium that has a refractive index . A beam of light traveling in air
shines perpendicularly on the glass. The beam contains blue light with
in air of 450 nm. What is the minimum non-zero thickness of the glass that gives
completely constructive interference for the blue light reflecting from the film?
5.5 Newton’s Rings
Newton’s Rings, first studied in 1717, is an interference pattern created between a convex
and a plane surface. A pattern of alternating dark and bright circles are seen when
monochromatic light is used. Dark rings are caused by destructive interference when light
waves are out of phase. Bright rings are caused by constructive interference when light
rays are in phase.
Consider a long focal length lens rests on a glass slab (figure below) so that a thin layer
of air is present between contact point of lens and glass slab. Let the thickness of air layer
is comparable to light wavelength.
Figure 5.4: Convex Lens placed on the glass slab
Let a beam of light is made incident in the normal direction to the surface of the lens. A part
of light reflects back from point B and a part of it passes through the air gap between lens and
glass surface and falls at point C. Here again a part of the beam is reflected back towards the
lens and a part is refracted in glass plate. If air gap between lens and plate is small, waves
reflected from points B and C form an interference pattern when combine. As light reflects
from points B and C in the same direction, then sum of the path and phase difference is
(5.5.1)
where is the refractive index of air and is the wavelength of the light being
used. Therefore for constructive interference (Bright zones)
76
(5.5.2)
And for destructive interference (dark zones)
(5.5.3)
Self Assessment Questions 1. How are Newton's rings formed?
2. Why is light reflected from upper side of lens and lower side of glass plate not
taken into account?
3. What is the logic of using the lens of larger radius of curvature in the experiment?
4. Why dark dot appears in the center of rings? Does it always exist?
5. Where is density of interference rings high and Why?
6. What type of interference pattern can be seen when white light is used in place of
monochromatic?
7. Is there any relationship between the number of formed rings and the lens’s radius?
8. Give some examples of application of interference.
5.6 Answers to Self Assessment Exercises
Answers to Self-Assessment Exercise 5.3 Q.1 ,
Q.2 (a) ,
(b) 3.00
Q.3 8:1
Q.5 m
Q.6 0.45 mm
Q.7 For Red mm and for Blue mm
Answers to Self-Assessment Exercise 5.4 Q.1 305 nm
Q.2 150 nm
References
1. Halliday, Resnik, and Krane, “ Fundamentals of Physics”, Wiley and Sons, Inc
2. Serway and Jewett. “Physics- A Calculus-Based Approach”, Books Cole in.
3. Sears, Zemansky and Young, “College Physics”, Amazon Publishers
4. Nelkon and Parker, “Advance Level Physics”, Hienemann Educational Books,
London.
77
UNIT-6
DIFFRACTION
Written by: Mukhtar Adarsh
Reviewed by: Dr. Amjad Imran
78
CONTENTS
Introduction ....................................................................................................... 79
Objectives ......................................................................................................... 79
6.1 Diffraction at Single Slit ....................................................................... 80
6.2 Intensity in a Single Slit diffraction Using Phasor Treatment .............. 80
6.3 Double Slit Interference and Diffraction Combined ............................. 81
6.4 Diffraction at a Circular Aperture ......................................................... 82
6.5 Diffraction at Multiple Slits .................................................................. 83
6.6 Diffraction Grating................................................................................ 84
6.7 Use of Spectrographs ............................................................................ 85
6.8 Dispersion and Resolving Power of Gratings ....................................... 85
6.9 Introduction to Holography................................................................... 86
References ......................................................................................................... 87
79
Introduction
In this chapter, we discuss about different features about diffraction and the wave theory
of light. We also discuss about the diffraction on multiple and single Slits and which
procedure is used in this mechanism and discuss about the intensity of light and their
diffraction pattern. The most important thing we discuss is their use in Spectrographs and
Holography that is newest concept.
Objectives
After completion of this unit you will be able to:
1. Define the terms used related to diffraction.
2. Describe how intensity of light affect diffraction.
3. Use spectrograph and holograph for solution of problems related to diffraction.
4. Use the techniques in photography and other life graphy.
80
6.1 Diffraction at Single Slit
When light pass through the narrow slit the light beams not only flare out far beyond the
geometric shadow of the slit they also give rise a series of alternating light and dark
resemble interference fringes is called diffraction or Diffraction pattern.
The simplest diffraction pattern to analyze is that produced by a long narrow slit. In this,
we discuss about maxima and minima location in pattern.
Consider a plane wave falling at normal incidence on a slit of width. Consider point P is
central .rays that is focus on point P, which are parallel to the horizontal axis. At this
condition, we first produced maxima.
Consider another point on the screen. Light rays that reach at point P1 leave the slit at
angle .The rays P1 passes undeflected through the centre of the lens and therefore
determine . Ray r1 originates at the top of the slit and ray r2 at its centre. In fact every ray
passing through the upper half of the slit there is corresponding ray passing through the
lower half, originates at point a/2 below the first ray, such that the two rays are out of the
phase at P1.So the condition for first minimum can be written
a/2 sin =
a sin θ = λ
Figure 6.1
6.2 Intensity in a Single Slit Diffraction Using Phasor Treatment
We had located the positions of the minima of the single-slit diffraction pattern. We now
wish to find an expression for the intensity of the entire pattern as function of the
diffraction angle θ. This expression will permit us to find the location and intensity of the
maxima. Before calculation, let us consider some qualitative conclusions about the
maxima and minima.
81
The figure shown below shows a slit of width a divided into N parallel strips, each of δx.
the strip are very narrow, so each strip cab be regarded as a radiator of Huygens wavelets
and all the light from a given strip arrives at point P with the same phase. The waves
arriving at from any pair of adjacent strips have the same (constant) phase difference ΔФ,
which can be found from
Phase difference/2π = path difference/λ
Or
ΔФ = 2π/λ δx sinθ
Figure 6.2
6.3 Double Slit Interference and Diffraction Combined
In our analysis of double slit interference, we assumed that the slits were arbitrarily
narrow—that is, that a << λ. For such narrow slits, the central part of the screen on which
the light falls is uniformly illuminated by the diffracted waves from each slit. When such
waves interfere, they produce interference fringes of uniform intensity.
For visible light, the condition a << λ is usually not met. For such relatively wide slits, the
intensity of the interference fringes formed on the screen is not uniform. Instead, the intensity
of the fringes varies within an envelope due to the diffraction pattern of a single slit.
The effect of diffraction on a double slit interference pattern is illustrated in the figure given,
which compares the double-slit pattern with the diffraction pattern produced by a single slit of
the same width as each of the double slits. In figure, the diffraction does indeed provide an
intensity envelope for the more closely spaced double slit interference fringes.
Let us now analyze the combined interference and diffraction pattern. The interference
pattern for two infinitesimally narrow slits is given;
=
Where
Β =
82
The intensity for the diffracted wave from either slit is given by equation
=
Where
We find the combined effect by regarding in equation as variable amplitude, given
in fact by of second equation. This assumption, for the combined pattern, leads to
=
Figure 6.3
6.4 Diffraction at a Circular Aperture
Other effects, including chromatic and spherical aberrations can distort the image formed
by a lens. These effects can be substantially reduced or eliminated by suitable shaping of
the lens surface or by introducing correcting elements into the optical system. However,
no amount of clever design can eliminate the effects of diffraction, which are determined
only by the size of the aperture and the wavelength of the light. In diffraction, nature
imposes a fundamental limitation on the precision of our instruments.
Figure 6.4
83
The mathematical analysis of diffraction by a circular aperture, which is beyond the level of
this text, shows that the first minimum occurs at an angle from the central axis given by
Sin θ = 1.22
Sin θ =
In the figure given below, the angular separation of the two point sources is such that the
central maximum of the diffraction pattern of one source falls on the first minimum of the
diffraction pattern of the other. This is called Rayleigh’s criterion for resolving images.
Two objects that are barely resolvable by Rayleigh’s criterion must have an angular
separation of
=
= 1.22
Figure 6.5
6.5 Diffraction at Multiple Slits
In the principle, we can use a double slit interferences pattern to measure the wavelength.
If we know the silt separation d, by locating the bright fringes on a screen and equation
dsin =m we find the wavelength. It is more convenient to determine by the separation
between adjacent maxima.
The interference pattern which was obtained by six slits offer a slight improvement note
that the bright fringes are narrower, meaning that we will be able to do slight better job of
measuring the fringe spacing and thus determine the wave length more precisely the
separation d between adjacent silts has the same value for both cases.
84
Figure 6.6
6.6 Diffraction Grating
Typical grating might contain N=10,000 slits distributed over a width of a few
centimeters, equivalent to a grating spacing D of few micrometers. Gratings are used to
determine the wavelengths and study the structure and intensity of the principal maxima.
A per iodic structure represent the diffraction grating for example a compact disk produce
a rainbow pattern when light is reflected from the disk surface. Grating produced their
images by transmit light. It is called Reflection gratings. Most grating used for visible
light whether it is reflected or transmitted type, are phase grating.
Grating are made by ruling equal spaced parallel grooves in a thin layer of gold or
aluminum deposited on a glass plate using a diamond cutting point whose motion is
automatically controlled by ruling engine.
If the grating were transparent, it could function as a transmission phase grating, if light
passes through different thickness they have different phases.
The simple grating can be used in spectroscope used for viewing the spectrum of a light
source, assumed to emit a number of discrete wavelengths.
85
6.7 Use of Spectrographs
1. It can be used in industrial research by infrared Spectrographic
2. It can be used in diffraction by diffraction spectrograph which measure the spectrum
3. It can be used to characterize plant biomass.
4. It can be used in astronomical spectrograph uses diffraction grating.
5. It is used in cure monitoring using optical fibers
6.8 Dispersion and Resolving Power of Gratings
6.8.1 Dispersion The ability of a grating to produce spectra that permit precise measurement of wave
lengths is determined by two intrinsic properties of grating
1. The separation between spectral lines that differ in wavelength by a small
amount .
2. The width or sharpness of the lines.
“The angular separation per unit wavelength interval is called diffraction D
of the grating.”
D=
Lines near equal wavelength appear as widely separated as possible
The physical property of grating is determined by its dispersion. The differential eq
d =md
in term of small differences
d =m
The dispersion D is given by
D=m/dcos
The dispersion increases as the spacing between the slits are decreases. The dispersion is
also increase by high order(m).
86
6.8.2 Resolving Power We obtain a reasonable measure of the ability to resolve nearby lines of different
wavelength by applying Rayleigh’s criterion. If the maximum of one line fall on the first
minimum of its neighbor we should be able to resolve the lines.
The limit of resolution of the grating occurs when two lines in the spectrum are separated
by a wavelength interval Such that the difference between angular positions. we can
define the resolving power R of the grating by
R=
If the lines are narrow then wavelength is small so the resolving power is large.
The physical propert of grating that determine R which give first minimum and
maximum.
m /dcos =
by solving R we get
R=Nm.
6.9 Introduction of Holography
The light emitted by an object contain the complete information on the size and shape of
the object. We can consider that information to be stored in the wave fronts of the light
from the object, specifically in the variation of intensity and phase of the electromagnetic
field. If we could record this information, we could reproduce a complete three
dimensional images of the object.
A Scheme for recording the intensity and phase of the waves from the object was developed
in 1948 By Dennis Gabor who was awarded the 1971 Nobel prize in physics for this
discovery .This type of image formation is called Holography , from the Greek words
meaning :entire picture” and the image is called Hologram. A wave diffracted from an object
interferes on the photographic film with a reference wave. Interference between two wave
serve as storing on the film information on the phase of the wave from the object.
Figure 6.9
87
Self Assessment Questions
1. When a monochromatic light is incident on a slit 0.222mm wide, the first
diffraction minimum is observed at an angle of 1.8 degree from the direction of the
incident beam. Find the wavelength of the incident light?
2. If u double the wide of single slit the intensity of the central maximum of the
diffraction pattern increases by a factor of four even through the energy passing
through the slits only doubles. Explain Quantitatively?
3. The two headlights of an approaching automobiles are 1.42mapart at what (A)
angular separation and (B) maximum distance will the eye resolve them?
4. Light of wavelength 440nm passes through a double slit , yielding the diffraction
pattern of intensity I versus deflection angle .calculate (a) the slit width and (b)
the slit separation (c) Verify Intensities of the m=1 and m=2 interferences fringes?
References
1. Volume 2 edition 5th physics Halliday-ResnickKrane
88
89
UNIT-7
INTRODUCTION TO POLARIZATION
Written by: Imran Amin Asim
Reviewed by: Arshad Mehmood Qamar
90
CONTENTS
Introduction ....................................................................................................... 91
Objectives ......................................................................................................... 91
7.1 Polarization ........................................................................................... 92
7.2 Production of Polarization .................................................................... 92
7.3 Polarization States ................................................................................. 94
7.4 Types of Polarization ............................................................................ 97
7.5 Specific Rotation of Plane of Polarization ............................................ 100
7.6 Uses of Polari Meters ............................................................................ 101
91
Introduction
Polarization is a fundamental property of light. While many optical applications are based
on systems that are “blind” to polarization, a very large number are not. Some
applications rely directly on polarization as a key measurement variable, such as those
based on how much an object depolarizes or rotates a polarized probe beam. For other
applications, variations due to polarization are a source of noise, and thus throughout the
system light must maintain a fixed state of polarization – or remain completely
depolarized – to eliminate these variations. And for applications based on interference of
non-parallel light beams, polarization greatly impacts contrast. As a result, for a large
number of applications control of polarization is just as critical as control of ray
propagation, diffraction, or the spectrum of the light. Yet despite its importance,
polarization is often considered a more esoteric property of light that is not so well
understood.
In this unit our aim is to answer some basic questions about the polarization of light,
including: what polarization is and how it is described, how it is controlled by optical
components, and when it matters in optical systems. Also we will discuss different types
of polarization and use of Polari-meter.
Objectives
After studying this Unit the students will be able to:
1. Define polarization.
2. Describe the polarization states.
3. Explain the types of polarization.
4. Understand the use of Polari-meter.
92
7.1 Polarization
Polarization generally just means “orientation.” It comes from the Greek word polos, for
the axis of a spinning globe. Polarization is a property that is common to all types of
vector waves. In classical physics, light is modeled as a sinusoidal electromagnetic wave
in which an oscillating electric field and an oscillating magnetic field propagate through
space. Since the magnetic field is always perpendicular to the electric field, we usually
sketch just the electric field when visualizing the optical wave's oscillations. Polarization
is defined in terms of the pattern traced out in the transverse plane by the electric field
vector as a function of time. Light is called natural or unpolarized if its plane of
polarization fluctuates randomly around the direction of light beam propagation, so that,
on average, no direction is favored. For example, most naturally produced light (sunlight,
firelight) is unpolarized. In any other case, the light beam can be considered to consist of
partially polarized or fully polarized light. The polarization of a light beam can be
represented by its electric field vector. Its optical power is a scalar quantity that is
proportional to the mean square of the electric field amplitude.
Light is an electromagnetic wave, i.e. a transverse sinusoidally varying electric field
whose direction, represented by vector E, is perpendicular to the direction of propagation
of the wave. If the orientation of E remains fixed as the wave moves forward, the light is
plane-polarized. The plane of polarization is the plane containing E and the direction of
propagation.
Natural or unpolarized light from an ordinary source such as a discharge lamp consists of
the total output from a very large number of randomly oriented atomic emitters. Each
excited atom radiates a linearly polarized wave train for ~ 10 -8
s and new wave trains are
constantly emitted so that the overall polarization changes in a completely random
fashion, equivalent to varying randomly on a 10 -8
s timescale.
7.2 Production of Polarization The most common method of polarization involves the use of a Polaroid filter. Polaroid
filters are made of a special material that is capable of blocking one of the two planes of
vibration of an electromagnetic wave. (Remember, the notion of two planes or directions
of vibration is merely a simplification that helps us to visualize the wavelike nature of the
electromagnetic wave.) In this sense, a Polaroid serves as a device that filters out one-half
of the vibrations upon transmission of the light through the filter. When unpolarized light
is transmitted through a Polaroid filter, it emerges with one-half the intensity and with
vibrations in a single plane; it emerges as polarized light.
A Polaroid filter is able to polarize light because of the chemical composition of the filter
material. The filter can be thought of as having long-chain molecules that are aligned
within the filter in the same direction. During the fabrication of the filter, the long-chain
molecules are stretched across the filter so that each molecule is (as much as possible)
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Figure 7.1
aligned in say the vertical direction. As unpolarized light strikes the filter, the portion of
the waves vibrating in the vertical direction are absorbed by the filter. The general rule is
that the electromagnetic vibrations that are in a direction parallel to the alignment of the
molecules are absorbed.
Figure 7.2
The alignment of these molecules gives the filter a polarization axis. This polarization
axis extends across the length of the filter and only allows vibrations of the
electromagnetic wave that are parallel to the axis to pass through. Any vibrations that are
perpendicular to the polarization axis are blocked by the filter. Thus, a Polaroid filter with
its long-chain molecules aligned horizontally will have a polarization axis aligned
vertically. Such a filter will block all horizontal vibrations and allow the vertical
vibrations to be transmitted (see diagram above). On the other hand, a Polaroid filter with
its long-chain molecules aligned vertically will have a polarization axis aligned
horizontally; this filter will block all vertical vibrations and allow the horizontal
vibrations to be transmitted.
Polarization of light by use of a Polaroid filter is often demonstrated through a variety of
demonstrations. Filters are used to look through and view objects. The filter does not
distort the shape or dimensions of the object; it merely serves to produce a dimmer image
of the object since one-half of the light is blocked as it passed through the filter. A pair of
filters is often placed back to back in order to view objects looking through two filters.
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By slowly rotating the second filter, an orientation can be found in which all the light
from an object is blocked and the object can no longer be seen when viewed through two
filters. What happened? In this demonstration, the light was polarized upon passage
through the first filter; perhaps only vertical vibrations were able to pass through. These
vertical vibrations were then blocked by the second filter since its polarization filter is
aligned in a horizontal direction. While you are unable to see the axes on the filter, you
will know when the axes are aligned perpendicular to each other because with this
orientation, all light is blocked. So by use of two filters, one can completely block all of
the light that is incident upon the set; this will only occur if the polarization axes are
rotated such that they are perpendicular to each other.
7.3 Polarization States
The polarization state is one of the fundamental characteristics that is required to study
light. There are three states of polarization that are used to describe light:
Linear Polarization
Circular Polarization
Elliptical Polarization
Linear polarization: If the polarization of all the electromagnetic waves in a light beam can be made so that
each of the electric or magnetic field vector to have the same orientation, then the light
beam is said to be polarized. Because of this, there is then a unique plane which contains
all the directions of the electric or magnetic field along with the light rays. This type of
polarization is referred to as plane polarization or linear polarization.
In electrodynamics, linear polarization or plane polarization of electromagnetic radiation
is a confinement of the electric field vector or magnetic field vector to a given plane
along the direction of propagation. See polarization and plane of polarization for more
information.
Figure 7.3
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The orientation of a linearly polarized electromagnetic wave is defined by the direction of
the electric field vector. For example, if the electric field vector is vertical (alternately up
and down as the wave travels) the radiation is said to be vertically polarized.
Linear polarizers can be divided into two general categories: absorptive polarizers, where
the unwanted polarization states are absorbed by the device, and beam-splitting
polarizers, where the unpolarized beam is split into two beams with opposite polarization
states. Polarizers which maintain the same axes of polarization with varying angles of
incidence are often called Cartesian polarizers, since the polarization vectors can be
described with simple Cartesian coordinates (for example, horizontal vs. vertical)
independent from the orientation of the polarizer surface. When the two polarization
states are relative to the direction of a surface (usually found with Fresnel reflection),
they are usually termed s and p. This distinction between Cartesian and s–p polarization
can be negligible in many cases, but it becomes significant for achieving high contrast
and with wide angular spreads of the incident light.
Circular Polarization The second polarization state is referred to as circular polarization. Circular polarization
can be described as the vectors of the electric field is rotated at a point in space in the
direction that is perpendicular to the plane of propagation, instead of fixed orientation
oscillation. The electric field vector magnitude also remains unchanged.
Circular polarization can be further classified according to the rotation. Looking at the
oncoming light wave, if the electric field vector of the light appears to be rotated in a
clockwise direction, then the wave is referred to as right-circularly polarized. On the
other hand, if the light vector appears to rotate in a counterclockwise direction, then the
wave is said to be left-circularly polarized.
Figure 7.4
A linear polarizing filter followed by a quarter-wave plate whose slow and fast axes are
at 45° to the axis of the polarizer becomes a circular polarizing filter, and incident
unpolarized light emerges as circularly polarized light. This will not work if the order of
the polarizer and wave plate is reversed. A quarter-wave plate converts circularly
polarized light into linearly polarized light.
96
The quarter-wave retardation plate is a sheet of birefringent (double refracting) material 1
of thickness such that horizontally and vertically polarized light entering in phase will
emerge from the retardation plate 1/4 of a wavelength out of phase. Unpolarized light is
not affected by this retardation plate (or by any thickness of birefringent material)
because the retardation plate only changes the phase of each component of polarization.
The situation dramatically changes when the incident light is polarized.
A polarizing filter is placed in front of the quarter-wave plate at a relative angle of 45° so
that the incident horizontal and vertical components are of equal intensity. Because of the
90° phase shift between the two components after they pass through the retardation plate,
the direction of polarization of the light that emerges from the wave plate will rotate in
time. Thus incident unpolarized light emerges as circularly polarized light. (More
generally, if the angle between the wave plate and polarizing filter is not 45°, the two
components will differ in intensity and the emerging light will be elliptically polarized.)
A second wave plate with the same orientation will result in a 180° phase shift, and the
components will now sum to obtain linearly polarized light that has been rotated by 90°.
The circular polarization produced by the linear polarizer/quarter-wave plate sandwich is
also made evident by placing a mirror behind it and looking through the circular polarizer
at the mirror reflection. The mirror reverses the direction of circular polarization, and the
reflected reversed circularly polarized light is converted back into linearly polarized light
by the wave plate. However, it is now polarized perpendicular to the linear polarizing
filter's orientation, so it is absorbed and the mirror appears dark 2 . The effect is undone
by rotating the linear polarizer with respect to the wave plate. The effect is also undone
by reversing the order of the polarizer and wave plate. Finally, one can substitute a non-
reversing mirror (two mirrors mounted together at right angles) to see what happens!
Elliptical Polarization The third state of polrization is called elliptical polariztion. Elliptically polarized light
consists of two light waves that are linearly polarized and having unequal amplitudes but
has the same frequency. This results in a light wave with electric vectors that both rotates
and changes its magnitude. An elliptical shape can be traced out by the tip of the electric
field vector, and therefore it is referred to as elliptical polarization.
It is also worthy to mention that circular polarization is a special case of elliptical
polarization. In electrodynamics, elliptical polarization is the polarization of
electromagnetic radiation such that the tip of the electric field vector describes an ellipse
in any fixed plane intersecting, and normal to, the direction of propagation. An
elliptically polarized wave may be resolved into two linearly polarized waves in phase
quadrature, with their polarization planes at right angles to each other. Since the electric
field can rotate clockwise or counterclockwise as it propagates, elliptically polarized
waves exhibit chirality.
Other forms of polarization, such as circular and linear polarization, can be considered to
be special cases of elliptical polarization. A beam of light is said to be elliptically
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polarized if the curve traced by the end point of the electric field vector is ellipse. This is
most general case of a polarized light. The equation of the ellipse can be obtained as
is an equation of a conic. It is obvious this conic section is confined in a rectangular
region with side parallel to the coordinate axes and whose lengths are 2Ax and2Ay.
All of the states of polarization described above are actually special cases of the most
general state of polarization, called elliptical polarization, in which the tip of the electric
field vector E traces out an ellipse in the x-y plane. The two components might have
unequal . That is, Ay , and also might contain a different relative phase, often denoted
amplitudes Ax we may write generally t kz xAx sinEx while the y
component is as before , y y sin kz t Ey A and where, as before, E = Ex + Ey.
The three special cases described in sections a, b, and c = 0 (linear polarization; equal
amplitudes); (b) Ay and above thus correspond to: (a) Ax /2 = – Ay with = 0
(linear polarization; unequal amplitudes); and (c) Ax Ay and Ax (circular
polarization). Some other examples of more general states of elliptical polarization are
shown below.
Figure 7.5
7.4 Types of Polarization
There are diffreents types of polarizations. Some of them are discuss below:
Plane of Polarization: Plane of polarization light is an electromagnetic wave, i.e. a transverse sinusoidally
varying electric field whose direction, represented by vector E, is perpendicular to the
direction of propagation of the wave. If the orientation of E remains fixed as the wave
moves forward, the light is plane-polarized. The plane of polarization is the plane
containing E and the direction of propagation natural or unpolarized light from an
ordinary source such as a discharge lamp consists of the total output from a very large
98
number of randomly oriented atomic emitters. Each excited atom radiates a linearly
polarized wave train for ~ 10 -8
s and new wave trains are constantly emitted so that the
overall polarization changes in a completely random fashion, equivalent to varying
randomly on a 10 -8
s timescale.
Figure 7.6
Polarization by Reflection: Your polarized light kit has five grayish filters, of which three are linear polarizers and
two are circular polarizers. In this part of the experiment, you should separate the two
different kinds of polarizers and mark the transmission axes on the linear polarizers.
You need a table with a lamp or a window near the far end. Put a magazine or a
book with a shiny cover on the table and put the microscope slide from the kit on
top. Look at an angle such that the reflection of the light makes it difficult or
impossible to read the print in the magazine and/or to read the print below the glass
microscope slide.
Figure 7.7 Figure 7.8
Now try each of the grayish filters in front of your eye. Rotate each filter to see if
you can find a position of the filter in which it will best reduce the glare. You can
then move the magazine and your head until the glare reduction with the filter is
most dramatic. Use that setup from now on. When you rotate the filter through 90o
the glare should become very bad again. Now flip the filter over so that you look
through the opposite side. Rotate the filter after flipping it over and try to find out if
it makes any difference whether you look through the filter from one side or the
other. For the linear polarizers, flipping should have no effect.
For circular polarizers, the behavior is distinctly different. Now you should be able
to separate your five filters into a group of three linear polarizers and two circular
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polarizers. List in your notes all the features you found that distinguish the two
groups
Hold each of the three linear polarizers in front of your eye so that it best reduces
the glare and then mark with a vertical line on small circular paper labels that you
can attach to each filter. Now you should be able to pick up each filter with the line
in a vertical position and get the best glare reduction. Your line marks the filter
transmission axis.
Now that you are a beginning expert, gather a variety of objects of different
materials; put them in place of the magazine and record what you observe with a
linear polarizer. Examples: a piece of paper on which you have written a few words
with a soft pencil, a shiny metal surface, the plastic box of your kit, and other
objects. Which objects shown polarization effects? The ability of polarizing filters
to preferentially block reflected light from water and other smooth surfaces makes
them valuable for sunglasses; they reduce glare far better than the unpolarized
variety.
Polarization by Scattering: On a clear day, if you look at the blue sky through a polarizer, you can turn the polarizer
so that the sky looks darker and white clouds stand out beautifully. Photographers use
polarizing filters for this purpose. The light from the sky is partially polarized, but you
must look in the right direction for a maximum effect. The blue color of the sky is due to
the fact that the short wavelengths (blue) of the light from the sun are scattered more
effectively by the atmosphere than the longer wavelengths (red).
Figure 7.9
In the following experiment, let light from an incandescent lamp pass through water with a
few drops of milk to scatter the light. If you look at the water through a linear polarizer, you
100
can see that the light intensity varies as you turn the polarizer. You can use this to find the
direction of polarization. The incident light is unpolarized, but the light scattered out of the
liquid is obviously partially polarized. The scattered light is produced when electrons in the
liquid are caused to oscillate transverse to the beam direction by the E-field of the incident
light. At a 90o scattering angle, these vibrations can only have components perpendicular to
the scattering plane or along the scattered direction. Since the E-field must vanish along the
propagation direction, the latter contribution must vanish. Verify that the polarization
direction is consistent with this explanation. The scattered light is only partially polarized
because part of the light is scattered more than once. In such multiple scattering, the light does
not remain in one plane. The line of sight is no longer always perpendicular to the direction of
propagation and multiply scattered light is less polarized.
7.5 Specific Rotation of Plane of Polarization
In chemistry, specific rotation (α) is a property of a chiral chemical compound. It is
defined as the change in orientation of monochromatic plane-polarized light, per unit
distance–concentration product, as the light passes through a sample of a compound in
solution. Compounds which rotate the plane of polarization of a beam of plane polarized
light clockwise are said to be dextrorotary, and correspond with positive specific rotation
values, while compounds which rotate the plane of polarization of plane polarized light
counterclockwise are said to be levorotary, and correspond with negative values. If a
compound is able to rotate the plane of polarization of plane-polarized light, it is said to
be “optically active”.
Specific rotation is an intensive property, distinguishing it from the more general
phenomenon of optical rotation. As such, the observed rotation (α) of a sample of a
compound can be used to quantify the enantiomeric excess of that compound, provided
that the specific rotation ([α]) for the enantiopure compound is known. The variance of
specific rotation with wavelength—a phenomenon known as optical rotatory
dispersion—can be used to find the absolute configuration of a molecule. The
concentration of bulk sugar solutions is sometimes determined by comparison of the
observed optical rotation with the known specific rotation.
Many crystals and dilutions have the ability to rotate the plane of the linearly polarized light
which is spreading through them. Such substances are called optically active. This property
comes from unsymmetrical structure of the molecule or the crystal lattice. Rotation of the
polarization plane occurs in the substances where crystal lattice or molecules cannot mirror
image of one match. Such objects are called chiral. Their asymmetry is resulting from a lack
of mirror symmetry. (Of everyday things have a chiral structure for example a bottle screw, as
well as a one hand glove. Optically active substances are quartz crystals, sugar, camphoric
and nicotine dilutions, etc. These substances usually occur in two different forms where the
crystal of one modification or a molecule is a mirror image of the other. One modification
turns then the light oscillation plane in one direction, another in the opposite direction. That is
the situation in case of quartz which rotates the light oscillation plane to the right and to the
left. The sugar molecule is asymmetric (screw shaped model) due to configuration of carbon
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atoms. There has been found 16 different forms of this molecule. They band together into 8
left-right pairs.
Rotation of light polarization plane in optically active substances can be explained by the
model of classical physics. According to that in optically active substance (screw-shaped)
molecules oscillating electric and magnetic dipoles are formed, which oscillations of the
field strength vectors are perpendicular to the initial (falling) field strength corresponding
vectors ( and ). Polarization plane turns as a result of accession of the initial and
induced electromagnetic fields. Phenomenological explanation to the phenomenon of the
optical activity was given by Fresnel in 19th century, based on the separation of the
polarized light into two in the opposite directions circularly polarized lights and assuming
different refractive indexes in case of the right and left circularly polarized light. Angle α
of the rotation of the polarization plane of the light transmitted through an optically active
substance is dependent on the layer thickness of the substance , the wavelength of light λ
, the temperature t and in case of dilutions, also on concentration c. It turns out that in
case of a given temperature and wavelength angle of the rotation of the polarization plane
α in dilutions is proportional to the thickness of the dilution and concentration of the
optically active substance in it:
∝= ∝ .
The physical quantity [∝] is called a specific rotation of an optically active substance. It
indicates the angle by which turns the polarization plane in case the light with wavelength
λ passes through a layer with a unit thickness and a unit concentration, the temperature of
which is t. In operational manuals specific rotation is usually given at wavelength of D-
line of sodium ( = 589.3nm) and at temperature = 20 °C. Corresponding specific
rotation is marked as ∝
Hence:
∝
shows that for designation of the specific rotation of the dilution it is necessary to know
the concentration of the dilution c, the thickness of the layer of dilution and the rotation
angle of the polarization plane α. In this work, dilution with a certain concentration, has
been poured into the cuvette cast. The length of the cuvette is written on it. The angle of
rotation is measured with the help of a polarimeter.
7.6 Uses of Polari Meters
Polari meters are analytical hardware components for measuring the polarization of light
and analyzing the optical activity of chemical substances. Optical activity refers to the
capability of a material to rotate beams of polarized light and the degree to which such
phenomena occur.
Manipulating the polarization of light through prisms or optical filters has been a
common practice for over 200 years, and these established principles are still largely
102
applied in modern polar meters. The instrumentation commonly comprises an incident
light source that is directed through a polarizing prism, which transmits light waves that
only fluctuate across a single plane. These polarized beams pass through a sample tube,
which can be loaded with an array of chemical solutions, before reaching a second prism.
By observing light transmitted through this analyzing prism, an observer can determine if
the sample has indeed rotated the light from the polarity predetermined by the initial
prism.
In other words, Polarimetry is an analytical method of measuring the concentration of
inorganic and organic compounds in a liquid solution using an instrument that determines
the optical activity of those compounds. A compound is considered to be optically active
if linearly polarized light is rotated when passed through it. The amount of optical
rotation is determined by the molecular structure and concentration of chiral molecules in
the substance. Each optically active substance has its own specific angle of rotation,
which can be used to identify specific compounds and their concentrations in a solution.
The polarimetry measurement does not affect the sample.
The polarimetry measurement is influenced by:
sample composition
concentration
length of the observation tube
temperature
color of light (wavelength)
Polarimetry is used in many industries.
Food Industries Polarimetry is used in food industries for quality control of original, intermediate, and
final products, the determination of concentrations, and purity control.
Sugar industries: sugar (sucrose, levulose, glucose, etc.), sugar syrups, starch,
sugar-free sweeteners like isomalt, etc.
Dairy: lactose, sucrose, lactoglobuline, lactic acid, esters, etc.
Vine industries: analysis of sugar on the vine, tartaric acid, esters, etc.
Fruit: analysis of sugar in fruit syrups (levulose), acids and esters (malic acid, etc.),
essential oils, etc.
Pharmaceutical Industries Polarimetry is used in pharmaceutical industries for purity control and determination of
concentration of substances according to the requirements of the European and American
Pharmacopeia by measurement of both specific and optical rotation.
Alkaloids: cocaine, codeine, nicotine, morphine sulphate, etc.
Amino acids: asparagine, glutamic acid, etc.
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Organic compounds: ascorbic acid, menthol, camphor, etc. Others: steroids,
antibiotics, serums, vitamins, etc.
Medicine
Research of sugar and albumin in urine
Hormone research
Enzymology and toxicology research
Cosmetic Industries Control of purity and identification of optically active essential oils and essences like
lemon oil, orange oil, lavender oil, spearmint oil, etc.
Chemical Industries Purity control and measurement of concentration, identification, and characterization of
compounds, such as:
Organic Fluids
Biopolymers
Synthetic polymers
Organic polymers
Research Applications
Analysis of optically active compound structure analysis
Determination of configuration changes of solved macromolecules
Monitoring changes in concentration of an optically active component in a reaction
mixture, as in enzymatic scission
Distinction of optical isomers
Investigating kinetic reactions by measuring optical rotation as a function of time
Analyzing molecular structure by determining optical rotation dispersion
Self Assessment Questions What is a light wave?
What is polarized light?
How is it possible to polarize the light?
What is a polaroid?
What is the intensity of the light?
What are the uses of polarimeter?
What is specific rotation?
Give the physical explanation to the rotation of the plane of polarization in an
optically active substance.
104
105
UNIT-8
ORIGIN OF QUANTUM THEORY
Written by: Imran Amin Asim
Reviewed by: Arshad Mehmood Qamar
106
CONTENTS
Introduction ....................................................................................................... 107
Objectives ......................................................................................................... 107
8.1 Black Body Radiation ........................................................................... 108
8.2 The Stefan-Boltzmann Law .................................................................. 109
8.3 Wien’s Displacement Law .................................................................... 110
8.4 Plank’s Law of Radiation...................................................................... 110
8.5 Quantization of Energy ......................................................................... 111
8.6 Correspondence Principle ..................................................................... 113
8.7 Einstein’s Photon Theory – The Compton Effect ................................. 114
8.8 Line Spectrum ....................................................................................... 116
107
Introduction
The origin of quantum theory is connected with a well-known phenomenon, which did
not belong to the central parts of atomic physics. Any piece of matter when it is heated
starts to glow, gets red hot and white hot at higher temperatures. The color does not
depend much on the surface of the material, and for a black body it depends solely on the
temperature. Therefore, the radiation emitted by such a black body at high temperatures
is a suitable object for physical research; it is a simple phenomenon that should find a
simple explanation in terms of the known laws for radiation and heat.
In this chapter we will discuss the characteristics of black body, the consequences of
Stefan, Boltzmann, Wien and plank’s laws. We will also discuss the Compton Effect and
Line Spectra with the help of Quantum theory.
Objectives
After studying this Unit the students will be able to:
1. Understand Black body Radiation.
2. Describe the consequences of Stefan, Boltzmann, Wien and Plank’s laws.
3. Explain the Compton Effect.
4. Elaborate Line Spectra with the help of Quantum theory
108
8.1 Black Body Radiation
A black body is an ideal body which allows the whole of the incident radiation pass into
itself and absorbs within itself this whole incident radiation this property is valid for
radiation corresponding to all wavelengths and all angles of incidene. Therefore the black
body is an ideal absorber of incident radiation. If a body is irradiated with radiation of
wavelength , and a fraction a( ) of that radiation is absorbed, the remainder being either
reflected or transmitted, a( ) is called the absorptance at wavelength . Note that l is
written in parentheses, to mean "at wavelength ", not as a subscript, which would mean
"per unit wavelength interval". The fractions of the radiation reflected and transmitted
are, respectively, the reflectance and the transmittance. The sum of the absorptance,
reflectance and transmittance is unity, unless you can think of anything else that might
happen to the radiation.
Consider two cavities at the same temperature. We'll suppose that the two cavities can be
connected by a "door" that can be opened or closed to allow or to deny the passage of
radiation between the cavities. We'll suppose that the walls of one cavity are bright and
shiny with an absorptance close to zero, and the walls of the other cavity are dull and
black with an absorptance close to unity. We'll also suppose that, because of the
difference in nature of the walls of the two cavities, the radiation density in one is greater
than in the other. Let us open the door for a moment. Radiation will flow in both
directions, but there will be a net flow of radiation from the high-radiation-density cavity
to the low-radiation-density cavity. As a consequence, the temperature of one cavity will
rise and the temperature of the other will fall. The (now) hotter cavity can then be used as
a source and the (now) colder cavity can be used as a sink in order to operate a heat
engine which can then do external work, such work, for example, to be used for
repeatedly opening and closing the door separating the two cavities. We have thus
constructed a perpetual motion machine that can continue to do work without the
expenditure of energy.
Figure: 8.1
109
Experience shows that the temperature of a hot and a cold object placed close to each
other equalize in vacuum as well. All macroscopic objects in all temperature emit (and
absorb) thermal radiation spontaneously. This radiation consists of electromagnetic
waves. The energy of the electromagnetic waves emitted by a surface, in unit time and in
unit area, depends on the nature of the surface and on its temperature. The thermal
radiation emitted by many ordinary objects can be approximated as blackbody radiation.
A perfectly insulated cavity that is in thermal equilibrium internally contains blackbody
radiation and will emit it through a hole made in its wall, provided the hole is small
enough to have negligible effect upon the equilibrium. The (absolute) blackbody absorbs
all energy, and reflects nothing, which is of course an idealization. A black-body at room
temperature appears black, as most of the energy it radiates is infra-red and cannot be
perceived by the human eye. Black-body radiation has a characteristic, continuous
frequency spectrum that depends only on the body's temperature. The spectrum is peaked
at a characteristic frequency that shifts to higher frequencies (shorter wavelengths) with
increasing temperature, and at room temperature most of the emission is in the infrared
region of the electromagnetic spectrum.
8.2 The Stefan-Boltzmann Law
The first quantitative conjecture based on experimental observations was the Stefan-
Boltzmann Law (1879) which states the total power radiated from one square meter of
black surface goes as the fourth power of the absolute temperature.
P=σT4
where
P is the total amount of radiation emitted by an object per square meter (Wattsm−2
)
σ is a constant called the Stefan-Boltzman constant (5.67×10−8
Wattsm−2
K−4
)
T is the absolute temperature of the object (in K)
The Stefan-Boltzmann Law is easily observed by comparing the integrated value (i.e.,
under the curves) of the experimental black-body radiation distribution at different
temperatures. In 1884, Boltzmann derived this T4 behavior from theory by applying
classical thermodynamic reasoning to a box filled with electromagnetic radiation, using
Maxwell’s equations to relate pressure to energy density. That is, the tiny amount of
energy coming out of the hole would of course have the same temperature dependence as
the radiation intensity inside.
The total power radiated from one square meter of black surface goes as the fourth
power of the absolute temperature.
Self Assessment Questions
What will be the effect on radiation power if the absolute temperature is increase
twice?
If the sun’s surface temperature is 5700 K. How much power is radiated by the
sun?
110
8.3 Wien’s Displacement Law
The second phenomenological observation from experiment was Wien’s Displacement
Law. Wien's law identifies the dominant (peak) wavelength, or color, of light coming
from a body at a given temperature. As the oven temperature varies, so does the
frequency at which the emitted radiation is most intense. In fact, that frequency is directly
proportional to the absolute temperature:
νmax∝T
where the proportionality constant is 5.879×1010
Hz/K.
Wien himself deduced this law theoretically in 1893, following Boltzmann’s
thermodynamic reasoning. It had previously been observed, at least semi-quantitatively,
by an American astronomer, Langley. This upward shift in νmax with T is familiar to
everyone—when an iron is heated in a fire, the first visible radiation (at around 900 K) is
deep red, the lowest frequency visible light. Further increase in T causes the color to
change to orange then yellow, and finally blue at very high temperatures (10,000 K or
more) for which the peak in radiation intensity has moved beyond the visible into the
ultraviolet.
Another representation of Wien's Law in terms of the peak wavelength of light is
λmax=b/T
where T is the absolute temperature in kelvin and b is a constant of proportionality called
Wien's displacement constant, equal to 2.89×10−3
mK, or more conveniently to obtain
wavelength in micrometers, b≈2900μm⋅K .This is an inverse relationship between
wavelength and temperature. So the higher the temperature, the shorter or smaller the
wavelength of the thermal radiation is. The lower the temperature, the longer or larger the
wavelength of the thermal radiation is. For visible radiation, hot objects emit bluer light
than cool objects.
Assessment If surface body temperature is 90 °F.
How much radiant energy in Wm–2 would your body emit?
What is the peak wavelength of emitted radiation?
8.4 Planck’s Law of Radiation
German physicist Max Plank successfully explained the energy distribution in the black
body radiation based on the following assumptions
The surface of the black body contains oscillators
These oscillators absorb or emit energy in terms of integral multiples of discrete
packets called quanta or photons. The energy ‘E’ of photons is proportional to the
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frequency ‘ν’ of the radiation. Mathematically E=nhν here h is a constant called
Planck’s constant and its value is 6.625 x 10 -34
Js, and ‘n’ can take integer values.
At thermal equilibrium the rate of absorption and emission of radiation are equal.
According to Planck’s law of radiation the expression for energy density of
radiation is given by
Where ‘c’ is the velocity of light, ‘k’ is
Boltzmann constant and ‘h’ is Planck’s
constant. This law explains the distribution of
energy in the black body radiation spectrum
completely for all wavelengths and at all
temperatures. Also this law can be reduced to
Wein’s distribution law in the shorter
wavelength region and to Rayleigh-Jeans law
in the longer wavelength region.
8.5 Quantization of Energy
The quantization of energy refers to the fact that at subatomic levels, energy is best
thought of as occuring in discreet "packets" called photons. Like paper money, photons
come in different denominations.
You can, for example, purchase items with a one dollar bill or a five dollar bill, but there
are no three dollar bills. Money, therefore, is quantized; it only comes in discreet
amounts.
In quatum physics, photons are packages of energy and correspond to different colors in
the spectrum or different types of electromagnetic radiation (radio waves, microwaves, X
-rays, etc).
A red photon has specific energy value different from a blue photon. The red and blue
photons are therefore "quantized" just as dollar bill denominations are "quantized". Each
photon contains a unique amount of discreet energy.
More technically, the uniqueness – or “quantization” – of energy is related to Plank's
constant, which specifies "how quantized" energy can get.
Figure 2: Plancks Radiation Graph
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The formula is: E=hf where E is energy, f is frequency and h is a very tiny constant called
the Plank's constant (6.62⋅10−36m2kgsec). This constant regulates and "quantizes" the
energy of the universe.
It might be helpful to mention some macroscopic analogies of this quantization of energy
phenomena. This is like a pendulum that has a characteristic oscillation frequency but can
swing with only certain amplitudes. Quantization of energy also resembles a standing
wave on a string that allows only particular harmonics described by integers. It is also
similar to going up and down a hill using discrete stair steps rather than being able to
move up and down a continuous slope. Your potential energy takes on discrete values as
you move from step to step.
Using the quantization of oscillators, Planck was able to correctly describe the
experimentally known shape of the blackbody spectrum. This was the first indication that
energy is sometimes quantized on a small scale and earned him the Nobel Prize in
Physics in 1918. Although Planck’s theory comes from observations of a macroscopic
object, its analysis is based on atoms and molecules. It was such a revolutionary
departure from classical physics that Planck himself was reluctant to accept his own idea
that energy states are not continuous. The general acceptance of Planck’s energy
quantization was greatly enhanced by Einstein’s explanation of the photoelectric effect
(discussed in the next section), which took energy quantization a step further. Planck was
fully involved in the development of both early quantum mechanics and relativity. He
quickly embraced Einstein’s special relativity, published in 1905, and in 1906 Planck was
the first to suggest the correct formula for relativistic momentum, p=γmv
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8.6 Correspondence Principle
The correspondence principle states that the any model describing the behavior of
quantum systems must yield the same results as classical physics in the macroscopic
limit.
This principle prevents quantum physicists from using models which contradict results
we know are true when we examine macroscopic systems. It helps us test whether a
model for quantum systems is correct.
Here is an example of the correspondence principle in action:
We know from quantum mechanics that matter can exhibit wavelike properties. The De
Broglie relation tells us if we have a particle with a momentum, pp, the wavelength of the
propagating matter wave is given by
λ=hp
where hh is Planck's constant. In macroscopic systems, the magnitude of the momentum
of particles is very large compared to hh. So therefore the wavelength of the particles in
macroscopic systems is zero, and we see no wavelike interference. This is why in your
day-to-day life you do not observe quantum interference.
The correspondence principle applies to other theories besides quantum theory. Thus the
mathematical formulations for the behavior of objects moving at exceedingly high
speeds, described by relativity physics, reduce for low values of speed to the correct
descriptions of the motions of daily experience.
The Bohr Correspondence Principle states that the predictions of quantum
mechanics must match the predictions of classical physics in the physical
situations that classical physics is intended to describe, and does describe very
accurately. Mathematically, this means that the equations of quantum mechanics
must smoothly turn into the equations of classical mechanics as the de Broglie
wavelength of particles becomes very small, and the energy state quantum
number gets very large.
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8.7 Einstein’s Photon Theory – The Compton Effect The Compton effect is the term used for
an unusual result observed when X-rays
are scattered on some materials. By
classical theory, when an electromagnetic
wave is scattered off atoms, the
wavelength of the scattered radiation is
expected to be the same as the wavelength
of the incident radiation. Contrary to this
prediction of classical physics,
observations show that when X-rays are
scattered off some materials, such as
graphite, the scattered X-rays have
different wavelengths from the
wavelength of the incident X-rays. This
classically unexplainable phenomenon
was studied experimentally by Arthur H.
Compton and his collaborators, and
Compton gave its explanation in 1923.
To explain the shift in wavelengths measured in the experiment, Compton used Einstein’s
idea of light as a particle. The Compton effect has a very important place in the history of
physics because it shows that electromagnetic radiation cannot be explained as a purely
wave phenomenon. The explanation of the Compton effect gave a convincing argument
to the physics community that electromagnetic waves can indeed behave like a stream of
photons, which placed the concept of a photon on firm ground.
Figure 8.4
The idea of the experiment is straightforward: Monochromatic X-rays with wavelength λ
are incident on a sample of graphite (the “target”), where they interact with atoms inside
the sample; they later emerge as scattered X-rays with wavelength λ′ . A detector placed
behind the target can measure the intensity of radiation scattered in any direction θ with
respect to the direction of the incident X-ray beam. This scattering angle, θ , is the angle
Figure 8.3
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between the direction of the scattered beam and the direction of the incident beam. In this
experiment, we know the intensity and the wavelength λ of the incoming (incident)
beam; and for a given scattering angle θ , we measure the intensity and the wavelength λ'
of the outgoing (scattered) beam. Typical results of these measurements are shown in
Figure 6.3.2 , where the x-axis is the wavelength of the scattered X-rays and the y-axis is
the intensity of the scattered X-rays, measured for different scattering angles (indicated
on the graphs). For all scattering angles (except for θ=0° ), we measure two intensity
peaks. One peak is located at the wavelength λ , which is the wavelength of the incident
beam. The other peak is located at some other wavelength, λ′ . The two peaks are
separated by Δλ ,, which depends on the scattering angle θ of the outgoing beam (in the
direction of observation). The separation Δλ is called the Compton shift.
Figure 8.5
8.7.1 Compton Shift As given by Compton, the explanation of the Compton shift is that in the target material,
graphite, valence electrons are loosely bound in the atoms and behave like free electrons.
Compton assumed that the incident X-ray radiation is a stream of photons. An incoming
photon in this stream collides with a valence electron in the graphite target. In the course
of this collision, the incoming photon transfers some part of its energy and momentum to
the target electron and leaves the scene as a scattered photon. This model explains in
qualitative terms why the scattered radiation has a longer wavelength than the incident
radiation. Put simply, a photon that has lost some of its energy emerges as a photon with
a lower frequency, or equivalently, with a longer wavelength. To show that his model
was correct, Compton used it to derive the expression for the Compton shift. In his
derivation, he assumed that both photon and electron are relativistic particles and that the
collision obeys two commonsense principles:
1. The conservation of linear momentum and
2. The conservation of total relativistic energy.
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8.8 Line Spectrum
Ordinary white light consists of waves of all wavelengths in the visible range. This is
why, when white light passes through a prism, a series of coloured bands are seen
called spectrum. This spectrum of white light ranges from violet at 7.5 x 1014
Hz to red
at 4 x 1014
Hz.
Since the colours merge into each other i.e. violet merges into blue, blue into green and
so on, we call it a continuous spectrum. When this light passes through an object or
medium, the wave with the shortest wavelength (violet) deviates the most than the one
with the longest wavelength (red).
Figure 8.6
Unlike visible light which shows a continuous spectrum of all wavelengths, the emission
spectra of atoms in the gas phase emit light only at specific wavelengths with dark spaces
between them. This is called line spectra or atomic spectra since the emitted radiation is
identified by bright lines in the spectra.
Line Spectra are the wavelengths of light emitted from an element when it's electrons
lose energy. The energy is lost in the form of photons of certain wavelengths, which are
different for each material. A Line Spectrum can be created by passing current through an
ionized gas. An Absorption Spectrum is produced by shining white light through a
sample of a gaseous element. The samples emission spectrum will be missing from the
result due to the absorption of energy by electrons corresponding to the energy levels of
the element. When current is passed through hydrogen gas, the hydrogen spectrum is
given as below:
Figure 8.7
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Line spectrum are unique for each element, and for each isotope of that element. An
energy level diagram shows the amounts of energy that electrons have at each level in an
atom. The energies are measured from a zero equivalent to a single free electron.
The hydrogen atom has the simplest line spectrum of all elements. For heavier atoms, the
line spectrum becomes more and more complex. However, there are certain features that
are common to all line spectra:
Line spectrum of every element is unique.
There is regularity in the line spectrum of each element.
Now, that we understand the line spectrum of hydrogen, let’s understand the features of
the hydrogen atom, its structure, and its spectrum.
Bohr’s model can explain the line spectrum of the hydrogen atom. According to
assumption 2, radiation is absorbed when an electron goes from orbit of lower energy to
higher energy; whereas radiation is emitted when it moves from higher to lower orbit.
Self Assessment Questions
Discuss black body radiation.
State and explain the Boltzmann’s Law.
What do you understand about Compton Effect?
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119
UNIT-9
THE WAVE NATURE OF MATTER
Written by: Imran Amin Asim
Reviewed by: Arshad Mehmood Qamar
120
CONTENTS
Introduction ....................................................................................................... 121
Objectives ......................................................................................................... 121
9.1 Wave Behavior of Particles .................................................................. 122
9.2 Wave Function and its Relation to Probability of Particle ................... 123
9.3 De Broglie Hypothesis and its Testing ................................................. 124
9.4 Davisson – Germer Experiment ............................................................ 125
9.5 Waves Packets and Particle .................................................................. 128
9.6 Localizing of Waves in Space and Time .............................................. 129
121
Introduction
What does wave nature of matter mean? Can a small particle be at multiple places at the
same time? Do I have a wave nature? Why can’t I see it? Let’s try to answer these
questions. we saw how light behaves both as a wave and particle. A particle is confined at
a place. On the other hand, a wave is spread in space. We say that the nature of light
depends on the nature of our observation. If you are observing phenomenon like the
interference, diffraction or reflection, you will find that light is a wave. However, if you
are looking at phenomena like the photoelectric effect, you will find that light has a
particle character.
You might ask, which is it? Is light a wave or a particle? The answer is that it has a dual
nature. You may also wonder whether it is a specific property of light! Does only light
have a dual nature? What if other quantities had dual nature? How could we measure and
prove that? Maybe these were the questions that led Louis Victor de Broglie to come up
with one of the most revolutionary equations in Physics, the de Broglie equation.
In this chapter we will discuss the concept of wave-particle duality, understand the wave
nature of matter, wave function and its relation to probability of particle, De Broglie
hypothesis and its testing, Devisson Germer Experiment and J.P Thomson Experiment,
Waves packets and particle and localization of waves in space and time.
Objectives
After studying this unit the students will be able to:
1. explain the concept of wave-particle duality,
2. understand the wave nature of matter
3. give an example of an application of wave-particle duality,
4. complete simple calculations using the de-Broglie wavelength equation
5. explain Devisson Germer Experiment and J.P Thomson Experiment.
6. localizing of waves in space and time.
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9.1 Wave Behavior of Particles
In the earlier articles, we saw how light behaves both as a wave and particle. A particle is
confined at a place. On the other hand, a wave is spread in space. We say that the nature
of light depends on the nature of our observation. If you are observing phenomenon like
the interference, diffraction or reflection, you will find that light is a wave. However, if
you are looking at phenomena like the photoelectric effect, you will find that light has a
particle character.
You might ask, which is it? Is light a wave or a particle? The answer is that it has a dual
nature. You may also wonder whether it is a specific property of light! Does only light
have a dual nature? What if other quantities had dual nature? How could we measure and
prove that? Maybe these were the questions that led Louis Victor de Broglie to come up
with one of the most revolutionary equations in Physics, the de Broglie equation.
Let us recall the mass-energy equivalence of Einstein, E =mc2 …(9.1)
Also from Einstein-Plank relation, we have: E = hν …(9.2)
Furthermore, we see that equation (1) is applicable to particles with some “mass”. In
other words equation (1) can be applied to particles and equation (2) is an equation for a
wave of frequency ν. So the two were not equated until de Broglie had a breakthrough!
We know that light can be a wave as well as a particle. In that case, we can say that
equation (1) and (2) represent the same quantity. Consequently, we must have: hν = mc2.
Since we know that ν = c/λ, we have:
h(c/λ) = mc2
λ = h/mc; where ‘c’ is the velocity of light. If we have a wave of velocity, say ‘v’, we can
write: λ = h/mv
or λ = h/p …(9.3)
where ‘p’ is the momentum of the wave-particle! See what we did here? We have mass –
a particle property, in the same equation as wavelength – a wave property. Thus if matter
exhibits wave properties, it must be given by equation (9.3). Equation (9.3) is the de
Broglie equation and represents the wave-particle duality. Hence we say that everything
in the Cosmos exhibits a dual nature. This is the wave nature of radiation and matter.
Let us find out your wavelength. Suppose you have a mass of 55 kg. If you are at rest i.e.
if the velocity = 0, then we see from equation (9.3), that λ is not defined. So not much
help there! Let us say that you are moving at a velocity of 5 m/s. Using equation (9.3), we
can see that
λ = h/(55)×5
λ = 6.63×10-34
/275 ≈ 2.4×10-36
m
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As you can see, you can’t “see” this small wavelength. Thus the wavelength of
macroscopic objects is too small to have any observable effects on any property at normal
velocities.
Self Assessment Questions What do you understand about wave nature of particle?
9.2 Wave Function and its Relation to Probability of Particle
A wave function is defined to be a function describing the probability of a particle's
quantum state as a function of position, momentum, time, and/or spin. Wave functions
are commonly denoted by the variable Ψ.
A wave function may be used to describe the probability of finding an electron within a
matter wave. To do this, the wave function, which may include an imaginary number, is
squared to yield a real number solution. Then, the probability of an electron being within
a certain area can be assessed.
A wave function in quantum mechanics is a description of the quantum state of a system.
The wave function is a complex-valued probability amplitude, and the probabilities for
the possible results of measurements made on the system can be derived from it. .The
state of such a particle is completely described by its wave function Ψ(x,t). So it’s the
Solutions of Schrodinger Equation
and the solution is Ψ(x,t) = A e^i(kx-wt) + B e^-i(kx-wt)
where x is position and t is time. This is a complex-valued function of two real variables
x and t. Following are the general forms of the wave function for systems in higher
dimensions and more particles, as well as including other degrees of freedom than
position coordinates or momentum components.
Since Ψ(x,t) is analogous to Amplitude of wave.
So for any kind of wave, we know I is directly proportional to A^2. The same is also for
Quantum Mechanics .Here the Intensity means frequently observing the particle in a
particular place in other words, Probability of finding the particle in a particular place.
Therefore chance or probability of finding the particle in place is obtained by |Ψ(x)|^2 =
Ψ(x)Ψ(x)* which is Probability Amplitude Function.
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Here given a graphical representation of Ψ(x) & Ψ(x)|^2 for an electron bounded in
different orbits in an atom to visualize the fact So, wave function is quantum state of any
object how it behaves in a system etc but probability density is that function by which we
determine the chances (probability) of finding that object in a certain time and space.
Self Assessment Questions Discuss Schrodinger Equation.
9.3 De Broglie Hypothesis and its Testing
The De Broglie hypothesis proposes that all matter exhibits wave-like properties and
relates the observed wavelength of matter to its momentum. After Albert Einstein's
photon theory became accepted, the question became whether this was true only for light
or whether material objects also exhibited wave-like behavior. Here is how the De
Broglie hypothesis was developed.
In his 1923 (or 1924, depending on the source) doctoral dissertation, the French physicist
Louis de Broglie made a bold assertion. Considering Einstein's relationship of
wavelength to momentum p, de Broglie proposed that this relationship would determine
the wavelength of any matter, in the relationship:
= h / p
recall that h is Planck's constant.
This wavelength is called the de Broglie wavelength. The reason he chose the momentum
equation over the energy equation is that it was unclear, with matter, whether E should be
total energy, kinetic energy, or total relativistic energy. For photons, they are all the
same, but not so for matter.
Assuming the momentum relationship, however, allowed the derivation of a similar de
Broglie relationship for frequency f using the kinetic energy Ek:
f = Ek / h
De Broglie's relationships are sometimes expressed in terms of Dirac's constant, h-bar = h
/ (2pi), and the angular frequency w and wavenumber k:
p = h-bar * kEk
= h-bar * w
The de Broglie hypothesis showed that wave-particle duality was not merely an aberrant
behavior of light, but rather was a fundamental principle exhibited by both radiation and
matter. As such, it becomes possible to use wave equations to describe material behavior,
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so long as one properly applies the de Broglie wavelength. This would prove crucial to
the development of quantum mechanics. It is now an integral part of the theory of atomic
structure and particle physics.
Young’s Double Slit Experiment
It is well-known that light has the ability to diffract around objects in its path, leading to
an interference pattern that is particular to the object. This is, in fact, how holography
works (the interference pattern is created by allowing the diffracted light to interfere with
the original beam so that the hologram can be viewed by shining the original beam on the
image). A simple illustration of light diffraction is the Young double slit experiment
Here, we use water waves (pictured as waves in a plane parallel to the double slit
apparatus) and observe what happens when they impinge on the slits. Each slit then
becomes a point source for spherical waves that subsequently interfere with each other,
giving rise to the light and dark fringes on the screen at the right.
Self Assessment Questions Discuss Schrodinger Equation.What is the relation between wavelength and momentum?
Explain young’s double Slit Experiment.
9.4 Davisson–Germer Experiment
Davisson and Germer Experiment, for the first time, proved the wave nature of electrons
and verified the de Broglie equation. de Broglie argued the dual nature of matter back in
1924, but it was only later that Davisson and Germer experiment verified the results. The
results established the first experimental proof of quantum mechanics. In this experiment,
we will study the scattering of electrons by a Ni crystal. Let’s find out more.
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The experimental setup for the Davisson and Germer experiment is enclosed within a
vacuum chamber. Thus the deflection and scattering of electrons by the medium are
prevented. The main parts of the experimental setup are as follows:
Electron gun: An electron gun is a Tungsten filament that emits electrons via thermionic
emission i.e. it emits electrons when heated to a particular temperature.
Electrostatic particle accelerator: Two opposite charged plates (positive and negative
plate) are used to accelerate the electrons at a known potential.
Collimator: The accelerator is enclosed within a cylinder that has a narrow passage for
the electrons along its axis. Its function is to render a narrow and straight (collimated)
beam of electrons ready for acceleration.
Target: The target is a Nickel crystal. The electron beam is fired normally on the Nickel
crystal. The crystal is placed such that it can be rotated about a fixed axis.
Detector: A detector is used to capture the scattered electrons from the Ni crystal. The
detector can be moved in a semicircular arc as shown in the diagram below.
Figure 9.1
The Thought Behind the Experimental Setup
The basic thought behind the Davisson and Germer experiment was that the waves
reflected from two different atomic layers of a Ni crystal will have a fixed phase
difference. After reflection, these waves will interfere either constructively or
destructively. Hence producing a diffraction pattern.
In the Davisson and Germer experiment waves were used in place of electrons. These
electrons formed a diffraction pattern. The dual nature of matter was thus verified. We
can relate the de Broglie equation and the Bragg’s law.
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From the de Broglie equation, we have:
where, m is the mass of an electron, e is the charge on an electron and h is the Plank’s
constant. Therefore for a given V, an electron will have a wavelength given by equation
above.
The following equation gives Bragg’s Law:
nλ = 2d sin(900- θ/2) …(9.4)
Since the value of d was already known from the X-ray diffraction experiments. Hence
for various values of θ, we can find the wavelength of the waves producing a diffraction
pattern from equation (9.4).
Observations of the Davisson and Germer Experiment The detector used here can only detect the presence of an electron in the form of a
particle. As a result, the detector receives the electrons in the form of an electronic
current. The intensity (strength) of this electronic current received by the detector and the
scattering angle is studied. We call this current as the electron intensity.
The intensity of the scattered electrons is not continuous. It shows a maximum and a
minimum value corresponding to the maxima and the minima of a diffraction pattern
produced by X-rays. It is studied from various angles of scattering and potential
difference. For a particular voltage (54V, say) the maximum scattering happens at a fixed
angle only ( 500 ).
Results of the Davisson and Germer Experiment From the Davisson and Germer experiment, we get a value for the scattering angle θ and
a corresponding value of the potential difference V at which the scattering of electrons is
maximum. Thus these two values from the data collected by Davisson and Germer, when
used in above equations give the same values for λ. Therefore, this establishes the de
Broglie’s wave-particle duality and verifies his equation as shown below:
As, we have:
For V = 54 V, we have
λ = 0.167 nm …. (9.5)
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Now the value of ‘d’ from X-ray scattering is 0.092 nm. Therefore for V = 54 V, the
angle of scattering is 500, using this in equation (9.3), we have:
nλ = 2 (0.092 nm)sin( 900−500/2)
For n = 1, we have:
λ = 0.165 nm ….. (9.6)
Therefore the experimental results are in a close agreement with the theoretical values got
from the de Broglie equation. The equations (9.5) and (9.6) verify the de Broglie
equation.
Self Assessment Questions Briefly discuss Davisson and Germer Experiment.
9.5 Waves Packets and Particle
We have seen how particles behave as waves in quantum mechanics. You've seen the
wave functions, and how a momentum eigenstate is a plane wave. But how do we
rationalize their particle-like behavior?
In one sense quantum particles are particle-like simply because they are countable, and
because the wave function is normalized. But how can we understand them moving in
classical trajectories? How do we describe a particle that is behaving in a sense that is in-
between the classical and quantum limits?
We describe such particles in terms of wave packets. A wave packet is a form of wave
function that has a well-defined position as well as momentum. Thus wave packets tend to
behave classically and are easy (and fun) to visualize. Naturally, neither the momentum nor
the position is precisely defined, as is governed by the uncertainty principle.
A classic application of the uncertainty principle is in understanding wave packets. A
wave packet with a very well-defined position will have a very uncertain momentum, and
thus will quickly disperse as the faster components move on ahead of the slower ones.
Conversely, if you construct a wave packet with a very definite momentum it will travel a
long distance without dispersing, but it starts out being very broad already in position
space.
The simples (and most commonly used) wave packet is a Gaussian wave packet. We can
construct such a wave packet most easily in reciprocal space:
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Waves and wave packets are mathematical concepts that happen to be very good at
predicting the behavior of simple particles like photons and electrons. Like those
particles, a wave can propagate from one place to another. And two waves can cancel
each other out in localized regions, matching what is observed when a beam of electrons
or light is made to pass through a pair of slits.
The wave with the simplest mathematical description is a plane wave, which has a single
wavelength and frequency, and which extends over all of space. Obviously any real
experiment involves particles that are more confined; a wave packet is something more
localized, corresponding to, for example, a pulse from a laser. It turns out you can
combine a bunch of plane waves of slightly different wavelengths and directions to form
a localized packet in any shape you like.
Even though a wave packet can maintain its confined shape and move through space, that
does not make it a particle. A wave that represents a particle has an additional property: It
is quantized. If you calculate how much of the wave there is (mathematically, you
integrate the 'norm' of the wave over all space), the answer you get is zero, one, or two,
etc. This reflects the fact that you can't have half of an electron or photon. That fact is
what makes those things particles.
9.6 Localizing of Waves in Space and Time
Wave packets can be formed by combinations of waves of specified frequency and
wavelength, and can have a range of characteristics. In the trivial case of constructing a
wave packet using only a single wave, we would as we have seen simply be describing an
object whose momentum is precisely specified but whose location is not specified at all.
At the opposite extreme, we could combine waves having different wavelengths so as to
create what amounts to a spatial delta function, thus describing an object that is precisely
localized; however, such a wave packet would need to include contributions from waves
of all possible wavelengths, so that this wave packet would represent an object whose
momentum would be quite indeterminate. Thus, neither extreme case would provide a
completely suitable representation of a classical object.
Figure 9.2
130
Let’s consider intermediate cases. A long wave train, limited in extent but long in
compared with the wavelength, has a relatively well-defined wave number in terms of its
Fourier transform, while a short wave train of limited extent would exhibit a broadened
distribution of wave numbers composing its Fourier transform.(2) In wave trains of even
shorter length in which the length of the wave train is comparable to or shorter than the
wavelength, we are no longer dealing with an ordinary wave train but rather with a wave
pulse whose principal wavelength component is comparable to the length or extent of the
sampling of the wave train. Wave packets exhibit a quite general feature that the degree
of localization of a wave packet in space making use of interference effects is inversely
correlated with the spread of available wave numbers. The unavoidable constraints on the
spatial extent and wave number content of a localized wave packet can be expressed in a
relationship that the product ∆x ∆k must be comparable to or greater than a number of the
order of unity, where ∆x is a measure of the spatial width of the wave packet and ∆k is a
measure of its spread in wave numbers. This relationship amounts to a fundamental
limitation on wave behavior, as fundamentally restrictive with respect to waves as, in
other contexts, the laws of thermodynamics are, or as the limiting speed of light is in
other physical systems. In quantum mechanics, these considerations are systematized and
the relationship of defined uncertainties is specified in terms of the more familiar form of
the Heisenberg uncertainty principle, which can be stated in terms of defined spatial and
momentum uncertainties by the inequality:
∆x∆p ≥ h/4π
Here ∆x and ∆p are the uncertainties in spatial location and in momentum respectively.
The Heisenberg uncertainty principle makes it clear that we cannot achieve a
simultaneous precise specification of location and momentum using a wave packet
representation to describe a classical object. However, we can form wave packets with
the minimum product of uncertainties allowed by the uncertainty principle; if we do so,
then we require that:
∆x∆p = h/4π
But, since p = h/λ, we find that, in magnitude:
∆p = h∆λ / λ2
Thus, the minimum uncertainty relation would correspond in magnitude to:
∆x∆λ = λ2 /4π
If we further specify that the uncertainty in wavelength and the uncertainty in x (both
spatial distances) should be comparable in magnitude, then we find that:
∆x ≈ ∆λ ≈ λ/(4π)½
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or, more roughly speaking:
∆x ≈ λ
Under these circumstances of wave packets exhibiting minimum uncertainties, the spatial
uncertainty in the location of a wave packet will be of the order of magnitude of a
wavelength. Alternatively considered, if we want the behavior of a wave packet to
resemble the behavior of a classical object, we need to minimize its spread in both space
and momentum space. To get near this goal, we typically need to require that the
momentum eigenfunctions participating in the wave packet should group around a central
momentum value characterizing the motion of the object. If a wave packet is formed with
positive momentum components with values centered around the central momentum, we
can form a relatively narrow packet using a range of values comparable to the central
momentum itself; if that approach is used, the wave packet width is typically of the order
of a wavelength corresponding to the central momentum value. Using either of these
approaches or comparable means of forming a semi-classical wave packet, we find that
wave packets that describe objects of reasonably well-defined velocity or momentum
typically have as a principal component a wave with a wavelength that corresponds
closely to the classical momentum of the object, and a width comparable to the
wavelength.
We have been describing spatial wave packets, but we could in a similar fashion consider
wave packets created in time, and lasting for a finite length of time, in connection with
describing quantum states of finite duration. Thus, for example, a laser pulse of finite
duration (in both space and time) could be described in terms of a wave train
characterized by a primary frequency; the Fourier transform of the pulse in frequency
would have a width dependent on both its frequency and the pulse duration and would
accordingly not be monochromatic.
Such temporal wave packets would necessarily exhibit a spread or width in energy, in
conjunction with the energy-time uncertainty principle that addresses the coupled
uncertainties in time and energy.
∆t ∆E ≥ h/4π
Self Assessment Questions 1. Explain Heisenberg uncertainly principle.
2. Discuss energy-time uncertainty principle.
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